My employer wants everyone in our unit to take the "Disc Classic Personality Profile Test System 2800" test, i.e. a shourt psychological test which identifies you as one of four personlaity types (or so it claims). Then we are all going to share our results and do team building exercises together.
Could this possibly be legal? Mandating a psychological test and then making a person share the results with everyone?
Any specific advise/reply would be greatly appreciated.

Could this possibly be legal? Yes.

Mandating a psychological test and then making a person share the results with everyone? Yes.

I doubt this is actually a "psychological test" in the sense you mean that phrase. It's not probing into your neuroses or your relationship with your mother. While I'm not familiar with this particular test, it's more likely a "styles" indicator - communication style, decision-making style, etc. Employers often use a styles indicator testing instrument (such as Myers-Briggs) as part of a team buiding exercise. The results can be quite interesting and sharing the results with the team and discussing them can be a lot of fun.

Relax and take the test. It's no big deal. If you want to know my Myers-Briggs profile, I'd be happy to tell you. :)