  We investigate the relation between pluri-Lagrangian hierarchies of
$2$-dimensional partial differential equations and their variational
symmetries. The aim is to generalize to the case of partial differential
equations the recent findings in [Petrera, Suris. J. Nonlinear Math. Phys.
24:sup1, 121--145 (2017)] for ordinary differential equations. We consider
hierarchies of $2$-dimensional Lagrangian PDEs (many of which have a natural
$(1+1)$-dimensional space-time interpretation) and show that if the flow of
each PDE is a variational symmetry of all others, then there exists a
pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time
Euler-Lagrange equations coincide with the original system supplied with
commuting evolutionary flows induced by the variational symmetries.
