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 (or the trader quoting two different prices for the same product would get fired)).
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?