One way to define the Steenrod Operations is to use the cup-i product, as in Mosher and Tangora's book. It basically says, given the chain complex from mod-2 homology $C\_\ast$, define

$D\_0 : C\_\ast\to C\_\ast\otimes C\_\ast$, 

so that the cup product is given (on cocycles)

$(u\cup v)(\sigma) = (u\otimes v)(D\_0\sigma)$

and then for higher $i$, define $D\_i$ so that 

$D\_{i-1}+\rho D\_{i-1} = D\_i\partial + \partial D\_i$

where $\rho$ is the flipping map. Then the $\cup\_i$ product is just

$(u\cup\_i u)(\sigma) = (u\otimes u)(D\_i\sigma)$

And then define for $[u]\in H^n$

$Sq^{2n-i}([u]) = [u\cup\_{i}u]$

This definition seems perfectly well-defined as a binary operation, and yet wherever I've seen it done it has only even been used as a unary operation. 

Is there a reason why this is the case, why either the product is undefined as a binary product or not useful as a binary product or just too hard to use?  

Is this a dumb question?

Thanks,
-Joseph