My understanding is that for some of the G10 currencies with negative rates (CHF, EUR), Swaption and Cap / Floor prices are quoted in terms of BOTH, normal and log-normal Vols. That in itself is not controversial, because these vols are self-consistent (you plug the quoted log-normal vol into the (shifted) log-normal formula: you get a specific price. You plug the normal vol into the normal option pricing formula: you must get the same price. Otherwise an identical option would have two different prices and someone would exploit the arbitrage.
What about the arising Greeks through and thereby hedging? I have even come across an implementation of the Libor market model which could switch between normal and shifted log-normal diffusion: don't these different models produce different Greeks? Intuitively, they shouldn't, but just looking at the basic Normal (Bachelier) vs. Log-normal (Black-Scholes) option pricing formulas, the Greeks will be different. Doesn't that imply that two different banks using two different models would calculate the risk differently, with one bank inherently miss-hedging?