  We study the one-dimensional (1D) quantum compass model with two independent
parameters by means of an exact mapping to the quantum Ising model. This allows
us to uncover hidden features of the quantum phase transition in the ordinary
one-parameter quantum compass model, showing that it occurs at a multicritical
point where a line of first-order transitions intersects a line of second-order
symmetry-breaking transitions of Ising type. We calculate the concurrence and
the block entanglement entropy in the four ground state phases, and find that
these entanglement measures accurately signal the second-order, but not the
first-order, transitions.
