  We study a generalization of the well-known classical two-dimensional square
lattice compass model of XY spins (sometimes referred to as the 90$^\circ$
compass model), which interpolates between the XY model and the compass model.
Our model possesses the combined $C_4$ lattice and spin rotation symmetry of
the compass model but is free of its fine-tuned subsystem symmetries. Using
both field theoretic arguments and Monte Carlo simulations, we find that our
model possesses a line of critical points with continuously varying exponents
of the Ashkin-Teller type terminating at the four-state Potts point. Further,
our Monte Carlo study uncovers that beyond the four-state Potts point, the line
of phase transition is connected to the lattice-nematic Ising phase transition
in the square lattice compass model through a region of first-order
transitions.
