  We prove that the total number $S_{n,132}(q)$ of copies of the pattern $q$ in
all 132-avoiding permutations of length $n$ is the same for $q=231$, $q=312$,
or $q=213$. We provide a combinatorial proof for this unexpected threefold
symmetry. We then significantly generalize this result to show an exponential
number of different pairs of patterns $q$ and $q'$ of length $k$ for which
$S_{n,132}(q)=S_{n,132}(q')$ and the equality is non-trivial.
