It's only weird if you posit that one must choose a side and then support that side in every issue. To me THAT is weird. And I'll scream as loudly as anyone if Obama proposes that labor have an equal share of control of a company as do its owners. That however doesn't mean that labor having ONE seat at the table (or more if labor chooses to invest in the company) isn't a good idea. For that matter, I think labor should have to invest in the company to get that seat, and in general I think the more labor invests in the company for which it works, the better for everyone involved. Worker-owned companies often take a longer view of success than companies run by "professional management", and the longer view is in my opinion better for our nation than is concentration on short-term profit.
And at the very least, labor having one free seat on the board would help the workers understand why management takes the actions it does, would possibly help management understand why some actions are bad ideas, and put a human face on decisions to export jobs or close down marginally profitable operations. If the BOD had to look a labor representative in the face, it might well decide to redirect or revise an unprofitable or marginally profitable operation rather than shut it down or outsource it, if only because they see the bad publicity they will get.