There has recently been renewed interest in the UV structure of loop amplitudes in perturbative N=8 supergravity in four dimensions. We explore the symmetries of prospective local counterterm operators in this theory by examining if their matrix elements satisfy the constraints of SUSY and R-symmetry Ward identities. This allow us to eliminate certain classes of operators as candidate counterterms. We impose further restrictions by studying compatibility with the spontaneously broken global E7(7)-symmetry of the theory; this is done with the help of soft-scalar theorems. The current status of N=8 UV finiteness/divergences will be summarized at the end of the talk.