  A class of Riemann-Cartan G\"odel-type space-times is examined by using the
equivalence problem techniques, as formulated by Fonseca-Neto et al. and
embodied in a suite of computer algebra programs called TCLASSI. A
coordinate-invariant description of the gravitational field for this class of
space-times is presented. It is also shown that these space-times can admit a
group $G_{r}$ of affine-isometric motions of dimensions $r=2, 4, 5$. The
necessary and sufficient conditions for space-time (ST) homogeneity of this
class of space-times are derived, extending previous works on G\"odel-type
space-times. The equivalence of space-times in the ST homogeneous subclass is
studied, recovering recent results under different premises. The results of the
limiting Riemannian case are also recovered.
