School of Seduction: The Circle of Silken Flowers (see: Platonic Prostitution, Band of Brothels) has its Velvet Academies (which concentrate just as hard on the arts of the salon and many other required topics, of course), and so forth. Rather classier, I suspect, than most of the examples given; after all, these are professional organizations with an emphasis on quality.
Band of Brothels: The Circle of Silken Flowers, as mentioned under Platonic Prostitution, along with a couple of its lesser cousins (including the more educational Guild of Erotic Artistry). In less platonic senses, the Negotiable Affection Division of the Eldinimieuthunimis runs a few of these abroad. (But even these latter enforce rather, ah, higher standards of professionalism and client selectivity than most.)
Platonic Prostitution: Played straightish for a variety of reasons: perhaps the primary of which is that the techlepathy (and in particular the telempathic component) that’s about to be mentioned when we get to Psychic Powers somewhat breaks the business model of the regular kind. Turns out basking in the less-than-warm glow of “I do the job, and then I get paid” is a kink shared by very few people.
That said, there are plenty of people who need a touch of elegance in their lives, someone interesting to talk to, some company in a lonely time or strange town, an escort for a society or professional event, or for that matter, someone to play that shiny new virtuality-game with at 3 am when everyone they know is busy. The Circle of Silken Flowers fills this role, and fills it admirably. (Which is not to say that it’s always or necessarily platonic; Imperial society doesn’t have any particularly obnoxious taboos in this area, and the Circle does largely recruit from people who find it easy to like people, and are picky about their clients, and so a not insignificant amount of the time events take a course which ends in a bed. But the sex, per se, is not what’s offered in the marketplace, nor a necessarily expected part of the service.)
(Which is also, just to be clear on the position, not to say that the Empire has laws, or any such thing, against offering sex on the open market either – it’s just that doing so, as mentioned above, tends to poison the goods, because you can’t both do it for money and not do it for money at the same time. Also, unlike the above, which is accepted in respectable society even if sex is involved, simply selling the sex directly for cash isn’t, in much the same way as “genuinely casual casual sex” isn’t – although not significantly more so. See Free-Love Future for more details there.)