I'm confused by the question and I think we need to clarify some details.
Seeing how there were harems, it made me wonder: Has there ever been a harem of all men, or at least if not, a male equivalent to it?
I'm not sure that I understand what OP means by the term harem - if the term is used correctly, it means
"a sacred inviolable place; harem; female members of the family")
Were there ever cultures that defined a sacred, inviolable place for male members of the family? Absolutely. There is a men's room in most public buildings, there used to be many male only clubs and organizations.
Tom Au interprets OP's question more broadly than I - "anything like it", and I think that answer is acceptable. But I don't see that within OP's question.
I am not a scholar of Islam or of the Ottoman empire, but my uneducated understanding is that what made the harem sacred and inviolable was the role of the harem in culture. Gender flipping that is a bit like dividing by zero - it makes no sense.
So how can we interpret OP's question?
- Did Ottoman culture have an institution which reserved certain spaces & customs for male only? Technically yes, but the sphere was the public sphere, and I think that is not in the spirit of the request.
In the Ottoman Turkish language, the harem, i.e., the part of the house reserved for women was called haremlik, while the space open for men was known as selamlık. Ibid
Did Ottoman culture have culturally recognized & supported customs concerning homosexual members of the family? Not my field, but I believe the answer is again technically yes, although I would be hesitant to describe them as "sacred". This is not my field, but I understand that the role of homosexuality in Ottoman culture was not dissimilar to that of many other cultures; homosexuality is permitted so long as it is not formally acknowledge. (that's a 1 sentence summary; reality is more complex, but I think that is sufficient for the context of this question)
Did Ottoman culture have an institution or space where women kept multiple lovers? I believe this is emphatically no. This is the typical meaning of "reverse harem". I don't have a firm date for the first occurrence of this term, but I'm confident that it is well after the end of the Ottoman empire.
Such a custom would have undermined the political dynastic role of the institution of the harem. In addition to "protecting" women and allowing them to develop a culture separate from the selamlik, the harem ensured that political wives were given adequate station and respect and ensured that the father of the harem offspring was undisputed.
male-equivalent harem only makes sense if you take the ( incorrect) Western Interpretation
A distinct, imaginary vision of the harem emerged in the West starting from the 17th century when Europeans became aware of Muslim harems housing numerous women. In contrast to the medieval European views, which conceived Muslim women as victimized but powerful through their charms and deceit. During the era of European colonialism, the "imaginary harem" came to represent what Orientalist scholars saw as an abased and subjugated status of women in the Islamic civilization. These notions served to cast the West as culturally superior and justify colonial enterprises. Under the influence of One Thousand and One Nights, the harem was often conceived as a personal brothel, where numerous women lounged in suggestive poses, directing their strong but oppressed sexuality toward a single man in a form of "competitive lust". _emphasis mine*
What would the function of such an institution be? The maternal lineage of any child is not obscure and does not need to be protected. I suppose that such an institution could reserve exclusive sexual privilege for the dominant female, but I'm not sure why that would be socially important. I suppose that it could serve to protect the status of foreign treaty consorts, but foreign treaty consorts seem unlikely without the potential for inheritance.
I believe it is a capital error to attempt to understand the role of a social institution without at least reference to the emic role of that institution. How did the Ottomans perceive the harem? what value did it have for them? How did they describe it. Attempting to extrapolate a "reverse harem" from a corrupt understanding of the institution will compound error on error.
Credit to Samid & Mark Johnson for making many of the same observations before I did, but I wanted to get the concepts out of comments and into an answer.