It is well-known (for example: chapter 2 in [GGM] A. Galligo, M. Granger, P. Maisonobe. D-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier Grenoble, 35 (1985), 1–48) that a perverse sheaf on $X=\mathbb{C}$ with stratification $X=\mathbb{C}^\times\sqcup \{0\}$ is equivalent to the datum of $(E,F,u,v)$, where $E$, $F$ are finite dimensional vector spaces, $u:E\to F$, $v:F\to E$ are linear maps such that $1+uv$ is invertible.

There is an obvious involution on the category of such data, namely $$(E,F,u,v)\mapsto (F,E,v,u).$$

My question is: what is the corresponding operation on perverse sheaves?

If I understand the correspondence correctly, the case $(E,F,u,v)=(\mathbb{C},0,0,0)$ corresponds to the constant sheaf $\underline{\mathbb {C}}_X$, and the case $(E,F,u,v)=(0,\mathbb{C},0,0)$ corresponds to the shifted sheaf $\underline{\mathbb{C}}_0[-1]$ supported at the origin. (I am using the convention of [GGM], so in particular the constant sheaf $\underline{\mathbb {C}}_X$ without shift is perverse.)

Is there any interesting interpretation of this involution?