It would still be the species but no longer the variety.
It would not really be a hybrid at all but it could be given a horticultural varietal name which is not the same as a botanical varietal name as such are given through a specific set of procedures and the submission of a paper detailing the defined variety.

If you cross any Cattleya intermedia var. coerulea with any other Cattleya intermedia (no matter what the variety) and any of the progeny have blueish lips they would be called variety coerulea.

You see many posts here where the word variety is used in reference to clones of hybrids. That is confusing.

If you are lucky enough to get an award with a species orchid you have to give the plant a clone name so that people know which particular plant was awarded.

Example. Cattleya intermedia variety amethystina 'Aranbeem' AM/AOS

There is no register of Clone names, so you can assign any name you like but you should avoid using names such 'Aranbeem', 'Carmela', 'H &R' and so on.

It kind of depends on whether you're talking about a variety in the botanical sense or whether your talking about the term variety as often misused by horticulturalists. Varieties are technically members of a distinct evolutionary lineage within a species that share a common ancestor. Therefore, intervarietal hybrids can't be considered to be either of the parental varieties in the same way that an interspecies hybrid can't be considered to be either of the parental species. ie Dendrobium speciosum var speciosum x Dendrobium speciosum var curvicaule is neither var speciosum nor var curvicaule regardless of how close the progeny look to either varieties.

You do have the problem where horticulturalists etc use the word variety to loosely refer to similar variations regardless of whether they share a common evolutionary lineage. The typical example is the use of latinised names for colour variants eg var alba. These names aren't usually accepted by taxonomists because the mutations that cause them can arise independantly or can be recreated by backcrossing and selfing outcrosses. ie just because a Paphiopedilum philippinense is green doesn't mean that it's from the same "alba" ancestor as the plant that was described as Paphiopedilum philippinense var album. In these cases, horticulturalists often will refer to a green/white plant produced by crossing a "var alba" with a red "alba" heterozygote as var alba but there was no validity in calling the parent plant var alba to begin with. The term group/cultivar group is probably a more appropriate designation for these varieties but orchid growers seem reluctant to make use of this horticultural rank. In saying that the RHS hasn't done the best job in publicising that they actually register groups.

You mention hybrid names. As far as hybrid registration goes, varieties, strains, subspecies, etc. are all lumped into the original species. Crosses with any varietal, form, strain or cultivar would all be called by the same hybrid name. It can get kind of confusing when alba and other color forms are used - you can end up with a white or yellow flower on a plant called "Purple Beauty" (lol) - but all in all, I think it'd be more confusing if every variation on a basic species got its own name for hybrid registration purposes, especially since sub-variations are often mislabeled.