We study twistor spinors (with torsion) on Riemannian spin manifolds carrying metric connections with totally skew-symmetric torsion. We consider the characteristic connection and under the condition , we show that the twistor equation with torsion w.r.t. the family can be viewed as a parallelism condition under a suitable connection on the bundle , where 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 implies that is both Einstein and -Einstein, in particular the equation holds for any . In fact, for -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 Kahler manifolds and nearly parallel -manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere . We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.