Working through Spivak's Calculus on Manifolds on page 79, is this a typo?
$$ \omega \wedge \eta = (-1)^{kl} \, \eta \wedge \omega $$
Working through the identity I'm getting instead: $$ \omega \wedge \eta = (-1)^{k+l} \, \eta \wedge \omega $$
If I'm wrong, can somebody show why?
There's a nice way to see that a permutation of the form $$ \binom{\,\color{blue}{1\,\ldots\,k}\;\color{green}{1\,\ldots\,l}\,} {\,\color{green}{1\,\ldots\,l}\;\color{blue}{1\,\ldots\,k}\,} $$ has length $kl$. Picture two highways running in parallel, one with $k$ lanes and the other with $l$ lanes. Now picture the roadway with $k$ lanes rising up and passing over to the other side of the roadway with $l$ lanes. Each of the $k$ lanes of one road must cross each of the $l$ lanes of the other and there are no other crossings. Counting these yields $kl$.
If you don't like highways, picture a braid with $k + l$ strands, where each of the $k$ strands must cross each of the $l$ other strands and there are no other crossings. Counting these crossings gives the length of the permutation (this number is minimal in this case), and reducing to parity gives the sign (doesn't depend on the crossings being minimal).
For example, with $k=3$ and $l=4$, we get $3 \cdot 4 = 12$ crossings, which is even, so the forms would commute without a sign. Here's a picture, where you can easily count the crossings.