  We introduce an exact spin transformation that maps frustrated
Z_{i,j}Z_{i,j+1} and X_{i,j}X_{i+1,j} spin interactions along the rows and
columns of the quantum compass model (QCM) on an LxL square lattice to
(L-1)x(L-1) quantum spin models with 2(L-1) classical spins. Using the symmetry
properties we unravel the hidden dimer order in the QCM, with equal two-dimer
correlations <X_{i,i}X_{i+1,i}X_{k,l}X_{k+1,l}> and
<X_{i,i}X_{i+1,i}X_{l,k}X_{l+1,k}> in the ground state, which is independent of
the actual interactions. This order coexists with Ising-type spin correlations
which decay with distance.
