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Relativistic gravitational anomalies lead to anomalous transport coefficients that can be activated at finite temperature in hydrodynamic and condensed matter systems with gapless, linearly dispersing fermions. One is the chiral vortical effect (CVE), an anomalous chiral current along the system's rotation axis, expressed in terms of a gravimagnetic metric field in a rotating frame with mixed gravitational anomaly. Another one arises in the presence of hydrodynamically independent frame fields (and spin connection) and leads to the thermal chiral torsional effect (CTE). We discuss the relation of CVE, CTE, and gravitational anomalies for relativistic fermions from the perspective of nonzero torsion and the Nieh-Yan anomaly when the currents depend on the frame fields and connection instead of the metric. The transport coefficients induced by the two gravitational anomalies at zero frequency and momentum are found to be closely related and equal. At level of linear response, their difference is demarcated whether or not torsion is nonzero and the existence of nonmetric degrees of freedom in the hydrodynamic constitutive relations with sources. In particular, the relativistic anomaly from torsion is well defined, since instead of an UV divergent term the chemical potential or temperature scales enter. This is closely related to the derivation of CVE from the fourth order in gradients gravitational anomaly and its appearance already in the linear response. However, the torsional anomaly is second order in gradients and directly contributes in linear response for CTE, implying also the same for CVE. For an example where the two anomalies are sourced independently, we consider chiral p+ip Weyl superfluids and superconductors rotating at finite temperature. At low energies in the linear approximation, the system is effectively relativistic along a special anisotropy axis. The hydrodynamics is governed by two velocities, the normal velocity vn and superfluid velocity vs. The existence of the two thermal anomalies in the condensate follows from the normal component rotation and the dependence of the momentum density on the superfluid velocity (order parameter). In the CVE, the chiral current is produced by the solid body rotation of the normal component with (angular) velocity vn=ω×r. In the CTE, a chiral current is produced by the vorticity of the superfluid velocity ∇×vs, which in the low-energy quasirelativistic effective theory plays the role of gravitational torsion. In thermal equilibrium, ⟪∇×vs⟫=2ω spatially averaged and the two gravitational anomaly currents cancel each other. This is a version of the Bloch theorem for axial currents, prohibiting finite current in equilibrium, realized as the cancellation of two gravitational anomalies with independent sources: gravimagnetic rotation field ω and torsion from vs. Although the latter physically represents the superfluid vorticity similar to the CVE, in the low-energy quasirelativistic theory, it arises from torsion coupling to the normal component chiral fermions.