  We consider a class of one-dimensional compass models with XYZ$-$YZX-type of
three-site exchange interaction in an external magnetic field. We present the
exact solution derived by means of Jordan-Wigner transformation, and study the
excitation gap, spin correlations, and establish the phase diagram. Besides the
canted antiferromagnetic and polarized phases, the three-site interactions
induce two distinct chiral phases, corresponding to gapless spinless-fermion
systems having two or four Fermi points. We find that the $z$ component of
scalar chirality operator can act as an order parameter for these chiral
phases. We also find that the thermodynamic quantities including the Wilson
ratio can characterize the liquid phases. Finally, a nontrivial magnetoelectric
effect is explored, and we show that the polarization can be manipulated by the
magnetic field in the absence of electric field.
