LeftPartialD[mu] denotes \overleftarrow{\partial }_{\mu } acting to the left.
Overview, ExpandPartialD, FCPartialD, LeftRightPartialD, RightPartialD.
QuantumField[A, LorentzIndex[\[Mu]]] . LeftPartialD[\[Nu]]
ex = ExpandPartialD[%]A_{\mu }.\overleftarrow{\partial }_{\nu }
\left.(\partial _{\nu }A_{\mu }\right)
ex // StandardForm
(*QuantumField[FCPartialD[LorentzIndex[\[Nu]]], A, LorentzIndex[\[Mu]]]*)StandardForm[LeftPartialD[\[Mu]]]
(*LeftPartialD[LorentzIndex[\[Mu]]]*)QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]] . LeftPartialD[\[Rho]]
ex = ExpandPartialD[%]A_{\mu }.A_{\nu }.\overleftarrow{\partial }_{\rho }
A_{\mu }.\left(\left.(\partial _{\rho }A_{\nu }\right)\right)+\left(\left.(\partial _{\rho }A_{\mu }\right)\right).A_{\nu }
ex // StandardForm
(*QuantumField[A, LorentzIndex[\[Mu]]] . QuantumField[FCPartialD[LorentzIndex[\[Rho]]], A, LorentzIndex[\[Nu]]] + QuantumField[FCPartialD[LorentzIndex[\[Rho]]], A, LorentzIndex[\[Mu]]] . QuantumField[A, LorentzIndex[\[Nu]]]*)