Context free languages invariant by "shuffling" right hand side

Given a grammar $G$ for a Context Free language $L$, we can augment it by "shuffling" the right hand side of each production, e.g.:

$A to BCD$ is expanded to $A to BCD ; | ; BDC ; | ; CBD ; | CDB ; | ; DBC ; | ; DCB$

It may happen that the resulting language $L’$ generated by the expanded $G’$ is equal to $L$

For example:

Source               Shuffled
S -> XA | YB         S -> XA | AX | YB | BY
A -> YS | SY         A -> YS | SY
B -> XS | SX         B -> XS | SX
X -> 1               X -> 1
Y -> 0               Y -> 0

Is there a name for such class of grammars ($L(G) = L(G_{text{shuffled}}))$?

Furthermore, does this theorem hold?

Theorem [it doesn’t hold, see Yuval’s answer and comments]: If exists a grammar $G$ such that, $L(G) = L(G_{text{shuffled}})$, then for all $G’$ such that $L(G) = L(G’)$ we have $L(G’) = L(G’_{text{shuffled}})$

In other words the "shuffle invariance" of grammars also corresponds to a class of languages.