In his 1937 ‘Monographic Study of the Red Crossbill’, Ludlow Griscom devoted 209 pages to the subject. Although he lacked field experience of old world taxa, he examined 546 specimens (compared with 2447 specimens from the new world). Amongst other findings, he considered that ‘specific’ characters between curvirostra (Common Crossbill) and pytyopsittacus (Parrot Crossbill) are completely bridged by the intermediate characters of crossbills in other geographic areas. He also considered that differences between the ‘white-winged’ crossbills were racial rather than specific (but see below). It is of note that he did not find any wing-barred (Common) Crossbills in the new world, whereas they are regularly recorded in the old world.
It is interesting to compare Griscom’s findings with a recent mtDNA-based phylogeny of crossbills (constructed by Laurent Raty) which can be found attached to post 200 here: http://www.birdforum.net/showthread.php?t=338168&page=8 The phylogeny does not show distinct genetic differences between curvirostra and pytyopsittacus (or for that matter scotica), though the sampled specimens of pytyopsittacus did not come from what is considered to be the core range of that taxon. It does however show distinct genetic differences between new world and old world (Common) Crossbills, and between bifasciata (Two-barred) and leucoptera (White-winged). It is not unusual for there to be genetic differences between new world and old world sister taxa, which are consequently treated as species rather than subspecies.
In our 2007 paper (https://britishbirds.co.uk/article/britains-first-two-barred-crossbill/ ), we described morphological and vocal differences between leucoptera and bifasciata which are now supported by genetic differences. It would therefore be reasonable to treat ‘White-winged’ and Two-barred’ Crossbills as species. They are certainly as or more distinct than other crossbill taxa currently treated as species.