Sorry about not posting links. You are correct about the Genesis Group. You can find some basic info about them at http://www.ldsgenesisgroup.org/ . Awhile ago I encountered an interview of their president, Darius Gray, describing some of their activities..

People might think that the Genesis group just meets in one group all by itself. But in fact, my understanding is you meet in the wards of where ever you are, but once a month you come together. Let’s clear up the misconception about how you meet, when you meet, etc.

We meet once a month. We are all active in our own wards, and we hold positions, many of us, in our own home wards in our stakes. Then we come together and share and celebrate who we are. We don’t have to lose our culture. We don’t have to lose our racial identity to be within the gospel.

We come together. It’s kind of like an LDS meeting that you might find in a sacrament meeting, but you don’t hear an “Amen” in an LDS sacrament meeting. But you can hear that at Genesis. There isn’t a pause after a musical number in a LDS sacrament meeting, and there generally is in Genesis. You generally don’t hear gospel or spirituals in an LDS meeting, and we do at Genesis. So, we’re a blend — we have LDS, we have our racial backgrounds, our cultural backgrounds, and they go together very well. You don’t have to have one exclusion of the other.

http://www.kued.org/productions/voices/interviews/gray.htm

The Genesis group is a recent example of such an auxillary. It may be the closest comparison to a religious order like Opus Dei. These auxillaries all target a specific demographic. Even if one is heavily involved in an auxillary they are still expected to be be involved in general church activities.

Some LDS communities historically attempted to live the Law of Consecration more fully; I would call them a religous order, but they were short lived and most of these were church initiated.

I would say grass roots organizations that stay independent, but unrecognized by the Church has exploded with the onset of the Internet. FAIR is a good example of this. Being active in FAIR is a definite lifestyle commitment, as any “FAIR widow” would attest to.

So I think the same underlying stimuli that would lead to a Catholic order being formed is present in Mormonism. But juggling regular church activities, parachurch(like FAIR) activities, and family responsibilities can be quite the strain. So a real religious order would be hard to pull off.