Any symmetric operad can be considered as an non-$\Sigma$ operad by throwing away permutations. Does anyone know what sort of structure one gets for algebras over $C_n$ the little n-cubes operad, or some equivalent variant, considered in this way? One thing which is strange is that there is a non-$\Sigma$ operad splitting of the map $Ass \to Com$ (where $Ass$ and $Com$ are the symmetric associative and commutative operads, respectively), and so any non-$\Sigma$ $C_n$ algebra should be a loop space, but I do not see what, if any, extra structure is present.
Justin, that is a cute question, and I've never thought about it. One starting thought free guess (acting on connected spaces) is that a non-$\Sigma$ $C_n$-space has $n$ possibly inequivalent but definitely related loop space structures. Do you see examples, or is this just curiosity? As to Ass and Com, I see no interest in building in the permutations to define Ass and then throwing them away. Of course, a non-$\Sigma$ `Com'-algebra is the same thing as an Ass-algebra, but the operadic modules over these kinds of algebras are different (bimodules and left modules). David, I think Justin is asking a concrete question, not the sort that model category structures shed light on.