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Consider a fixed set of gates of the kind $\land_n(P_{\varphi})$. With $P_{\varphi}$ being a relative phase shift gate by $\varphi$.
Can I assert that any permutation of that set is equivalent?
Example:
Can I rearrange this circuit at my liking and always get the same evolution?
Yes, as long as you change the order of the gates without changing the qubits on which each gate acts.
Proof: All phase gates are represented by diagonal unitary matrices. If $A$ has a diagonal matrix then controlled-$A$ also has a diagonal matrix. Therefore, all gates in your set of gates have diagonal matrices. Conclusion follows from the fact that diagonal matrices commute.
