  Torsion is a non-Riemannian geometrical extension of general relativity that
allows including the spin of matter and the twisting of spacetime. Cosmological
models with torsion have been considered in the literature to solve problems of
either the very early (high redshift $z$) or the present-day Universe. This
paper focuses on distinguishable observational signatures of torsion that could
not be otherwise explained with a scalar field in pseudo-Riemannian geometry.
We show that when torsion is present, the cosmic duality relation between the
angular diameter distance, $D_{\mathrm A}$, and the luminosity distance,
$D_{\mathrm L}$, is broken. We show how the deviation described by the
parameter $\eta = D_{\mathrm L}/[ D_{\mathrm A}(1+z)^2] -1 $ is linked to
torsion and how different forms of torsion lead to special-case
parametrisations of $\eta$, including $\eta_0 z$, $\eta_0 z/(1+z)$, and $\eta_0
\ln (1+z)$. We also show that the effects of torsion could be visible in
low-redshift data, inducing biases in supernovae-based $H_0$ measurements. We
also show that torsion can impact the Clarkson-Bassett-Lu (CBL) function ${\cal
C}(z) = 1 + H^2 (D D'' - D'^2) + H H' D D'$, where $D$ is the transverse
comoving distance. If $D$ is inferred from the luminosity distance, then, in
general non-zero torsion models, ${\cal C}(z) \ne 0$. For pseudo-Riemannian
geometry, the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric has ${\cal
C}(z) \equiv 0$; thus, measurement of the CBL function could provide another
diagnostic of torsion.
