Voiculescu’s freeness emerges when computing the asymptotic spectra of polynomials on [Formula: see text] random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix [Formula: see text]. In this paper, we elaborate on the previous result by proposing a random matrix model, which we name the Vortex model, where [Formula: see text] has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector [Formula: see text]. In the limit [Formula: see text], we show that [Formula: see text] matrices randomly rotated by the matrix [Formula: see text] are asymptotically conditionally free with respect to the normalized trace and the state vector [Formula: see text]. We define a new concept called cyclic-conditional freeness “unifying” three independences: infinitesimal freeness, cyclic-monotone independence and cyclic-Boolean independence. Infinitesimal distributions in the Vortex model can be computed thanks to this new independence. Finally, we elaborate on the Vortex model in order to build random matrix models for [Formula: see text]-freeness and for [Formula: see text]-freeness (formerly named indented independence and ordered freeness).