  We present here some connections between the liberation process for
projections $(P,Q)\mapsto(P,U_tQU_t^*)$ and its counterpart
$(R,S)\mapsto(R,U_tSU_t^*)$ for symmetries when the projections $\{P,Q\}$ and
the symmetries $\{R,S\}$ are associated, where $U_t$ is a free unitary Brownian
motion freely independent from $\{P,Q\}$ (and so $\{R,S\}$). We relate the
moments of their actions on the operators $X_t:=PU_tQU_t^*$ and
$Y_t:=U_tRU_t^*S$ and use this to prove a relationship between the
corresponding spectral measures (hereafter $\mu_t$ and $\nu_t$). On the other
hand, we focus in the process of unitary random variables $Y_t$ in the case of
arbitrary trace values $\tau(R),\tau(S)$. More precisely, we use stochastic
calculus to derive a partial differential equation (PDE for short) for its
Herglotz transform and use it to develop subordination results in terms of
L\"owner equations. The paper is closed with an improved proof of $i^*\left(
\mathbb{C}P+\mathbb{C}(I-P); \mathbb{C}Q+\mathbb{C}(I-Q)
\right)=-\chi_{orb}\left(P,Q\right)$ as an application.
