  We use exact symmetry properties of the two-dimensional quantum compass model
to derive nonequivalent invariant subspaces in the energy spectra of $L\times
L$ clusters up to L=6. The symmetry allows one to reduce the original $L\times
L$ compass cluster to the $(L-1)\times (L-1)$ one with modified interactions.
This step is crucial and enables: (i) exact diagonalization of the $6\times 6$
quantum compass cluster, and (ii) finding the specific heat for clusters up to
L=6, with two characteristic energy scales. We investigate the properties of
the ground state and the first excited states and present extrapolation of the
excitation energy with increasing system size. Our analysis provides physical
insights into the nature of nematic order realized in the quantum compass model
at finite temperature. We suggest that the quantum phase transition at the
isotropic interaction point is second order with some admixture of the
discontinuous transition, as indicated by the entropy, the overlap between two
types of nematic order (on horizontal and vertical bonds) and the existence of
the critical exponent. Extrapolation of the specific heat to the $L\to\infty$
limit suggests the classical nature of the quantum compass model and high
degeneracy of the ground state with nematic order.
