I understand that, based on market convention, you can construct the volatility surface of an indirect quote FX pair by flipping the volatility surface of the direct quote around the ATM level. For example, the 25 Delta Call volatility of USD/JPY is the same as the 25 Delta Put volatility of JPY/USD.
I am wondering if this is an approximation, though, because I have tried deriving the strikes of (1) USD/JPY call option that gives you 25% delta and (2) JPY/USD put option that gives you -25% delta, but they are not exactly inverse of each other.
Any insight on this matter is greatly appreciated!