  We present a general formulation to analyze the structure of slowly rotating
relativistic stars in a broad class of scalar-tensor theories with disformal
coupling to matter. Our approach includes theories with generalized kinetic
terms, generic scalar field potentials and contains theories with conformal
coupling as particular limits. In order to investigate how the disformal
coupling affects the structure of relativistic stars, we propose a minimal
model of a massless scalar-tensor theory and investigate in detail how the
disformal coupling affects the spontaneous scalarization of slowly rotating
neutron stars. We show that for negative values of the disformal coupling
parameter between the scalar field and matter, scalarization can be suppressed,
while for large positive values of the disformal coupling parameter stellar
models cannot be obtained. This allows us to put a mild upper bound on this
parameter. We also show that these properties can be qualitatively understood
by linearizing the scalar field equation of motion in the background of a
general-relativistic incompressible star. To address the intrinsic degeneracy
between uncertainties in the equation-of-state of neutron stars and
gravitational theory, we also show the existence of universal
equation-of-state-independent relations between the moment of inertia and
compactness of neutron stars in this theory. We show that in a certain range of
the theory's parameter space the universal relation largely deviates from that
of general relativity, allowing, in principle, to probe the existence of
spontaneous scalarization with future observations.
