In short: the Corda crossfeed is more neutral than the Headroom one. However, the Headroom crossfeed is more “fun” to listen to.
A while back, I auditioned the HA-2 and compared the Corda crossfeed to my headroom Supreme’s crossfeed. The Corda is very subtle. On some music, I could hardly tell the difference between crossfeed (medium processing) and no crossfeed.
The difference between having the crossfeed on/off on the headroom is much more apparent. The image shifts “up” more on the headroom and there seems to be more “blending” of the right and left channels. The downside to the headroom is that the crossfeed messes up vocals a bit. Even with the high pass filter turned on to compensate for the “duller” processed sound, vocals still seems a bit more weighted.
My suggestion would be: if you have a thing for vocals (bossa nova, jazz, opera, etc.), then the Corda crossfeed would be more suitable. However, if you are more into having a “fun” listen (by this I don’t mean that the Headroom is non-audiophile) then the Headroom gives more kick to the music.
Of course, the amps sections might have played a part in the sonic characteristics described. Personally, I preferred the headrom "house" sound.