Was there ever a male-equivalent to a harem in the Ottoman Empire?

There would not have been a male equivalent of an all male harem in the Ottoman Empire or elsewhere.

Harems were set up to serve the interests of powerful male rulers, and also to ensure that these male rulers would produce heirs. Almost never in history was there any powerful female rulers to be "served" by such an arrangement.

Possibly the one exception to the above was Queen Elizabeth I. But even though she was the queen, there were severe restrictions on her sexuality. In fact, she had the reputation of being the "Virgin Queen."

The reason is because of the different roles of men and women in the reproductive process. A man can sleep with any number of women, and know which partner is the parent of which child. A woman (in the days before DNA tests) could not sleep with multiple men and know for sure which man was the father of her child, unless she slept with only one man. And if royal power is conferred based on birth, people need to feel that the birth is "legitimate."

The consequences of a queen sleeping with multiple men could destroy her kingdom, which is why Queen Elizabeth I remained virginal. Her cousin {Mary Queen of Scots]1 did sleep with "too many" men and lost her throne and head as a result.(Mary lost her throne after marrying Bothwell, who was suspected of murdering her earlier husband Darnley, and was executed after a (platonic) correspondence with Anthony Babington.) Call this a double standard if you will, but queens were held to a much higher standard than kings because of their maternal function.

Edit: One can argue that Empress Catherine the Great of Russia kept an "unofficial" harem (of lovers). But in a world where men had most of the power, no queen would be allowed to keep a harem formally.

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?

1. 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.[10] Ibid

2. Did Ottoman culture have culturally recognized & supported customs concerning h*m*sexual 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 h*m*sexuality in Ottoman culture was not dissimilar to that of many other cultures; h*m*sexuality 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)

3. 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.

[Due to the ambiguities of the wording, it's unclear if you mean to limit yourself to only the Ottoman empire. If so, then I'll delete this answer.]

If you're strictly looking for an Ottoman equivalent, then I believe you're out of luck. If you mean whether there's ever been an equivalent to a harem, you have a few different examples from Chinese and Roman history.

Wu Zetian

After the death of her husband, Emperor Gaozong 高宗, the seventh-century Wu Zetian 武則天 called herself emperor, created her own dynasty, and, if the sources can be believed, had a series of at least four male favorites.¹

Later authors likely exaggerated the size of her harem, but at least the four (including the infamous Zhang brothers) are rooted in history.

Liu Chuyu

From Wikipedia:

After her father's death in July 464, her full younger brother Liu Ziye became emperor (as Emperor Qianfei). She became one of the people who often attended him while he visited places outside the palace. On one occasion, she told him:

"While our genders are different, we are born of the same father. However, you have more than 10,000 women in your palaces, and I only have one husband, and this is unfair."

In response, Emperor Qianfei selected 30 young handsome men for her, calling them her mianshou (面首, literally meaning "prime faces"), for them to be her lovers. From this point on in Chinese history, mianshou became a term for women's male lovers, often referring to lovers of honored women. He also promoted her to the greater title of Princess Kuaiji.

Roman Homosexuality

You also have the Roman case of exoleti, a type of Roman male prostitute. From Wikipedia again

Ancient sources impute the love of, or the preference for, exoleti (using this or equivalent terms) to various figures of Roman history, such as the tribune Clodius, the emperors Tiberius, Galba, Titus, and Elagabalus.

It's implied that Elagabalus, for example, had multiple male favorites.