When we value interest rate derivatives on any date $t$, we can estimate our future payments using some calibrated forward curve $f_s$, where $s$ is the spot date, and discount these back to $t$ using some calibrated discounting curve $d_t$.
Some software vendors (such as Bloomberg) consider any payments between $t$ and $s$ as being historical cash flows, which basically means that we value the swap on the spot date, not today. This approach has the benefit that any spot-starting swap can be directly compared to any legacy swap, since only the payments that come after the spot date will be considered (take for instance the case of a spot-starting swap vs a swap traded 1 year ago - the latter will have a payment on the spot date, this yielding present value, dv01 and par rate calculations different among the two swaps if we do not ignore the payment on the spot date). The drawback of this approach is that we naturally are not valuing the swap today which can be a problem for MTM on trading books.
Is there any convention for handling such accrued payment issues? Do we ignore all cash flows up until and including the spot date, such that we can directly compare, for instance, par rates of legacy swaps to spot-staring swaps, or is the convention to have a preference for "correct" date valuation?