Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, etc.). This category is also still a monoidal category (the opposite of a monoidal category is canonically a monoidal category). How do all of these possible adjectives interact? Is it still closed? Does the push-out product axiom still hold? It seems like we must lose some things, for instance, if we wanted the axiom "for any cofibrant object $X$ there is an equivalence $1\otimes X\simeq X$" this can't in general be true anymore, can it, since the cofibrant objects are completely different now?

In general it seems clear that things break down, but I'm interested in precisely what breaks down. Do we still have a simplicial model category, but we're just missing some of the monoidal model category axioms? I'm also okay with dumping the simplicial requirement.