We study twistor spinors (with torsion) on Riemannian spin manifolds
$$(M^{n}, g, T)$$
carrying metric connections with totally skew-symmetric torsion. We consider the characteristic connection
$$\nabla ^{c}=\nabla ^{g}+\frac{1}{2}T$$
and under the condition
$$\nabla ^{c}T=0$$
, we show that the twistor equation with torsion w.r.t. the family
$$\nabla ^{s}=\nabla ^{g}+2sT$$
can be viewed as a parallelism condition under a suitable connection on the bundle
$$\Sigma \oplus \Sigma $$
, where
$$\Sigma $$
is the associated spinor bundle. Consequently, we prove that a twistor spinor with torsion has isolated zero points. Next we study a special class of twistor spinors with torsion, namely these which are *T*-eigenspinors and parallel under the characteristic connection; we show that the existence of such a spinor for some
$$s\ne 1/4$$
implies that
$$(M^{n}, g, T)$$
is both Einstein and
$$\nabla ^{c}$$
-Einstein, in particular the equation
$${{\mathrm{Ric}}}^{s}=\frac{{{\mathrm{Scal}}}^{s}}{n}g$$
holds for any
$$s\in \mathbb {R}$$
. In fact, for
$$\nabla ^{c}$$
-parallel spinors we provide a correspondence between the Killing spinor equation with torsion and the Riemannian Killing spinor equation. This allows us to describe 1-parameter families of non-trivial Killing spinors with torsion on nearly Kähler manifolds and nearly parallel
$${{\mathrm{G}}}_2$$
-manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere
$${{\mathrm{S}}}^{3}$$
. We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.