For equity and fx vol surfaces you'd generally prefer to interpolate linearly in [$]\sigma^2t[$] over [$]\sigma[$] (keeping the forward moneyness constant) to avoid calendar arbitrage. For swaption or caplet vol interpolation it's not so clear why [$]\sigma^2t[$] is the better choice, since the underlying changes with [$]t[$] and hence there is no arbitrage argument supporting the decision.
Are there other reasons why you'd still interpolate IR Vols in [$]\sigma^2t[$] rather than [$]\sigma[$]. And what do people / systems actually do? Always assuming you do not something completely different like interpolating SABR parameters.