  We develop a three-dimensional $\mathcal{N}=4$ theory of rigid supersymmetry
describing the dynamics of a set of hypermultiplets
$(\Lambda^{\alpha\alpha'\dot{\alpha}'}_I,\,\phi^{\alpha A}_I)$ on a curved
AdS$_3$ worldvolume background, whose supersymmetry is captured by the
supergroup ${\rm D}^2(2,1;\, \boldsymbol{\alpha})$. To unveil some remarkable
features of this model, we perform two twists, involving the SL$(2,\mathbb R)$
factors of the theory. After the first twist, our spacetime Lagrangian exhibits
a Chern-Simons term associated with an odd one-form field, together with a
fermionic "gauge-fixing'', in the spirit of the Rozansky-Witten model. The
second twist allows to interpret the ${\rm D}^2(2,1;\, \boldsymbol{\alpha})$
setup as a framework capable of describing massive Dirac particles. In
particular, this yields a generalisation of the Alvarez-Valenzuela-Zanelli
model of ''unconventional supersymmetry''. We comment on specific values of the
combination $\alpha+1$, which in our model is related to a sort of gauging in
the absence of dynamical gauge fields.
