You asked about the relationship between Hazrat-e Mevlana and Hz. Shamsu 'd-din of Tabriz.
First of all, it is necessary to understand that in Persian sufi poetry, the word "lover" [`âshiq] means being a lover of God. And in the paths of sufism that view the mystic seeker as the lover and God as the Beloved, it means a true dervish. Therefore, "the lovers" are the lovers of God. So in this sense Mevlana and Shams certainly were "(spiritual) lovers."
Next, its necessary to consider how much the words "love" and "lover" have become sexualized in the English language. Only thirty years ago, for example, "making love" in popular songs meant no more than hugging and kissing. Now it always means "having sexual relations." Similarly, "lovers" now always means "people who have or had sex together." There is no longer any concept of lovers who don't engage in sex with each other: such as "unrequited lovers," meaning people in love who are unable to be sexual; or "Platonic lovers," who are in love, but choose not to have a physical relationship; or "spiritual lovers," such as the celibate Catholic nuns who view themselves as "married" to Christ. Next, it is necessary to recall how much homosexuality has increasingly become accepted and viewed as natural in our culture. As a result, it is more common to think/assume/suspect that men who are exceptionally close to each other and enjoy spending time together might be homosexuals or bisexuals.
As a result, when we read that when Mevlana and Shams first met, they were so enthralled with each other that they spent several months secluded together [actually, in the prayer-retreat cell of Mevlana's disciple, Husamuddin]. For the Western reader, the thought is almost irresistible to wonder if they might have had a sexual, as well as a deeply spiritual, relationship. After all, we know how sexual energy builds up over time, and they were so happy to be together, etc. Andrew Harvey, an openly gay author of books on Rumi, is said to have proclaimed this in public lectures as a fact (at least in lectures he gave while on the faculty of a private graduate school in San Francisco during the 1990's; see also about his "Teachings").
And distorted versions of Rumi's poetry (not only his) are largely responsible for giving a false impression of "Rumi's sensual side," such as references to "nudism"-- in which he is depicted as becoming so ecstatic that he would tear of all his clothes. (But public nudity is forbidden in Islam and this "tearing" was done by dervishes during samâ` [sema] and involved tearing one's cloak [khirqa] into pieces, or tearing the upper part of one's shirt or -- something done in a symbolic way in the Mevlevi samâ` when the shaykh turns in the center while holding the cloak as if just "ripped" from the collar to the lower chest.)
However, there is no evidence of a "physical relationship" between these two great sufi saints, and it is a suspicion or assumption with no basis. And it is a also Western misunderstanding of Persian poetry and Persian culture in the context of Islam and Islamic mysticism. In Islamic societies there has been a general segregation of men and women for over a thousand years. As a result, men are closer to each other than we can readily understand-- and they are so without being any more homosexual (in a religion that strongly condemns it). When my wife and I were in Istanbul many years ago (1977), it was common to see pairs of men walking and holding hands (but this custom had nothing at all to do with homosexuality). Yet after the markets closed, and there were no other women on the streets in a conservative neighborhood, and my wife felt insecure about this and held my hand, there seemed to be many disapproving stares--because (as we were later told) it is discouraged for men and women to hold hands in public in places where conservative Muslims are the majority (however, such areas are much diminished in today's very cosmopolitan city of Istanbul).
In terms of traditional themes and imagery in Persian sufi poetry, it is very common for the beloved to be praised as having beautiful tresses of hair, eyes, cheeks, moles, eyebrows, etc. And when Mevlana used such images in his poems expressing his spiritual love for Shams, this can be mistakenly interpreted as some kind of "evidence" of homosexual love. However, this was a centuries-old convention in Persian poetry that was long adopted by sufis who understood the various imagery in praise of the beloved as symbols of mystical love.
In the context of Islam, Mevlana and Shams were both very pious Muslims. Mevlana was a religious authority who inherited the mantle of religious scholarly authority from his father. He also earned income to support his family as an Islamic teacher and judge. He was a Sunni Muslim who followed the Hanafî school of Islamic law. We have more information about Shams now, from his "Discourses" [Maqâlât], a collection of excerpts from his talks written down by his disciples. We know that he was not an uneducated, "wild", or "heretical" dervish. He was a Sunni Muslim, with a solid Islamic education in the Arabic language, who followed the Shâfi`î school of Islamic law. There are translated quotes from Shams in which he criticized other sufi teachers as "not following" the example of the Prophet sufficiently. We know that Mevlana was married during the time he knew Shams. And we know that Mevlana arranged for Shams to marry a young woman raised in Mevlana's household, Kîmiyâ (= "Alchemy").
It is also helpful to understand their relationship in terms of the sufi teaching of the stages of "passing away" or "annihilation" [fanâ]. In this particular sufi path, the disciple is encouraged to cultivate love for the spiritual master within the heart, to visualize the master in the heart or seated in front of one, and to remember the master frequently. This practice is said to lead to mystical experiences of seeing the spiritual master (or "beloved") everywhere and the master's beauty expressed in all things waking or dreaming). Mevlana seems indeed to have been in this type of "passing away in the spiritual presence of the master [fanâ fî-sh- shayk], because he wrote thousands of verses expressing his spiritual love for Shams in his Divan. Part of this particular teaching is that if this closeness with the spiritual master [shaykh] goes on too long, it can become a barrier to "annihilation in God" [fanâ fî 'llâh]. And Shams suggested directly to Mevlana that he might have to go away for Mevlana to progress further. After Shams disappeared permanently, and after Mevlana recovered from his loss, it is said that Mevlana found Shams in his own heart. And in his last years, Mevlana composed thousands of couplets (the Mathnawi) in which he describes many unitive mystical experiences (usually spoken by one of the characters in a story), and rarely mentions the name of Shams. This is very much like "annihilation in God" following "annihilation in the master."
Also, in the context of mystical Islam, the dervishes obviously loved to spend time with each other, doing ritual prayer, zikru 'llâh, etc. Although Islam strictly condemns homosexual behavior, yet homosexual relations would occur sometimes between men and adolescent boys, due to the segregation of unmarried males together (and this continues down to the present day, as described in a recent news report about the revival of this centuries-old practice in the Afghan city of Kandahar). Mevlana condemns homoxexuality among dervishes (see below). And Mevlana, Shams, and Mevlana's father have all been quoted as condemning a practice engaged in by some sufis involving homoerotic gazing at attractive young adolescent boys (a type of "Platonic love" in which the gazer contemplates Divine Beauty in a lovely "beardless youth").
Mevlana condemns sodomy and effeminate behavior in numerous places in the Mathnawi. He said, "The (true) sufi [Sûfî] is the one who becomes a seeker of purity [Safwat]; not from (wearing) garments of wool [Sûf] and sewing (patches) and sodomy. With these vile people, sufism [Sûfiyî] has become stitching and sodomy [al-liwâTa] and that is all" (V:363-64). This contrast between purity and sodomy would appear to echo one of the passages in the Qur'an which mentions the Divine punishment of the people to whom the Prophet Lot was sent. When he confronted them ("Would you commit this abomination with you eyes open? Must you approach men with lust instead of women?"), they responded with sarcasm by urging that Lot and his followers be expelled, "For they are a people who would stay pure" (Qur'an 27:54-58). There are five references in the Mathnawi to the fate of the people of Lot. (The Arabic word used in Persian for sodomy, "liwâTa," is derived from these same "people of Lot," or "LûT" in Arabic.)
In a related story, a group of attractive women chide a man, saying that despite the multitude of women, men "fall into sodomy [liwâTa] because of the (supposed) scarcity of women" (VI: 1727-32). See also the story of the eunuch and the homosexual (V: 2487-2500), and the story of the beardless youth who tried to protect himself from a homosexual in a sufi gathering place (VI:3843-68). Most illegal sexual behavior of this kind occurred between men and "beardless youths," a behavior which Mevlana clearly condemns. Also, in Aflâkî's book of stories (completed 70 years after Mevlana died), according to one account: "Similarly, when HaZrat-é Mawlânâ made (his son) Walad the disciple [murîd] of Mawlânâ Shamsu 'd-dîn Tabrîzî-- may God sanctify their spirits-- he declared, 'My Bahâ' ud-dîn (Walad) doesn't consume hashish and never engages in sodomy [liwâTa], since, before God the Most Bountiful, these two behaviors are greatly disapproved and blameworthy.'" (Manâqibu 'l-`ârifîn, IV:32 (see also the translation by John O'Kane, "The Feats of the Knowers of God," p. 436).
Professor Franklin Lewis has given an excellent rebuttal to Western fantasies of the relationship between Mevlana and Shams in his excellent book (which recently won an award), "Rumi-- Past and Present, East and West: The Life, Teachings and Poetry of Jalâl al-Din Rumi," 2000, in his section "Modern Myths and Misunderstandings," pp. 317-326. He points out that Mevlana was about 37 when he met Shams, and that according to Mevlevi tradition Shams was 60 years old. He described how the homoeroticism in the Persian culture of Mevlana's time was very different from the homosexuality in ours. The penetrated boy held a socially inferior status. "A stigma attached to being penetrated, and a self-respecting mature male would not allow this to happen to himself." [Lewis, p. 322] A dominant male, who had been attracted to androgynous boys also desired women and would eventually marry and have children. "When a boy passed a certain age and grew facial hair, he himself became a member of the sexually dominant class and would no longer submit to penetration. Violation of these social norms led to scandal and legal prosecution. [Lewis, p. 323]
"The suggestion that the relationship between Shams and Rumi was a physical and homosexual one entirely misunderstands the context. Rumi, as a forty-year-old man engaged in ascetic practices and teaching Islamic law, to say nothing of his obsession with following the example of the Prophet, would not have submitted to the penetration of the sixty-year-old Shams, who was, in any case, like Rumi, committed to following the Prophet and opposed to the worship of God through human beauty. Rumi did employ the symbolism of homoerotic, or more properly, androgynous love, in his poems addressed to Shams as the divine beloved, but this merely adopts an already 300 year-old convention of the poetry of praise in Persian literature." [Lewis, p. 324]
If, having read the following up to this point, your mind continues to nag you with suspicious questions about Rumi and Shams spending long periods of time together, read something that Hazrat-e Shamsu 'ddîn-e Tabrîzî himself said about this (as recorded by his disciples):
"(Regarding) me and Mawlânâ, if (the time for the ritual prayer) becomes lost for us, without (our) intending (it during) a time of being occupied, we are discontent because of that and we make up (the missed prayer) alone (together). And when I don't go (on) the day of Jum`ah (the day of the Friday congregational prayers), there is sadness for me, (feeling) that, "Why didn't I join that (gathering) with this spiritual reality [within me]? Although it is not real distress, yet it is (there)." --from "Maqâlât-e Shams-e Tabrîzî," pp. 742-43 (see William Chittick's translation of selections of this important work of Shams' "Discourses," not previously available in English, "Me and Rumi: the Autobiography of Shams-i Tabrizi," (Fons Vitae, 2004, p. 80). [NOTE: The interpretation in parentheses, "(the time for the ritual prayer)" is correct because the word translated as "make up (the missed prayer)" [qaZâ] is a technical term in Islam for a ritual prayer that is missed during one of the five daily prayer times and is done afterwards.]
"The intended aim [maqSûd] of the world's existence is the encounter of two friends (of God) [mulâqât-e dô dôst bow-ad], when they face each other (only) for the sake of God [jehat-e khodâ], far distant from lust and craving [dûr az hawâ]. The purpose is not (for) bread, soup with bread crumbs, butcher, or the butcher's business. It is such a moment as this, when I am tranquil in the presence of Mawlânâ [ba khidmat-e mawlânâ âsûda'êm]."
--from "Maqâlât-e Shams-e Tabrîzî," p. 628 (see Refik Algan's translation of selections from this work, from Turkish with final Englishing by Camille Helminski, "Rumi's Sun: The Teachings of Shams of Tabriz", 2008, pp. 269-70.)
Hope this helps to resolve popular misconceptions about the relationship between Hazrat-e Mevlana and Hz. Shams. As Mevlevis, we should defend the honor of the great saints of God [awliyâ 'ullâh] such as these, who were the founders of our tradition. (And may Allah forgive us for our own suspicions!)