Abstract  A class of Riemann-Cartan Gödel-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 Gödel-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.