In a comment below, TSO makes a suggestion along a line of thought I haven't much considered:
It seems to publically identify oneself with a [political] party or ideology is to some extent compromise the mission, at least if you are a member of a religious order.
I don't publicly identify myself with a party or ideology because I don't privately identify myself with a party of ideology. Now that I think of it, though, I'm not at all confident I'd do well identifying the party affiliation (if any) of the Third Order (much less First Order) Dominicans I know. (The party affiliation of the Second Order seems generally to be Ice Cream.)
If anyone else has any thoughts about what doors open or close with respect to apostolic works as a result of a person belonging to a political party or identifying with an ideological camp, I'd love to hear them.
And here's a question to the Reader: To what extent does the American phenomenon of belonging to a political party without a strong sense of association with that party (e.g., how many members of either major party have ever attended a party meeting?) affect the American perspective on one's religious affiliation?