Finally got some more time on the LCX last night. I think it's still evolving some. I originally didn't find the LCX and HD6xx to be the best of matches. But that's changed and I'm sadly not sure if it's because I've grown used to it or that the amp is still burning in? I have less that 25 hours on the thing so far...so not a ton of time. Although I'm used to needing to burn-in tubes which is of course not a consideration here.
Anyway..on to my thoughts (standard disclaimer, I have relatively older ears (45+ yrs) that have been abused by loud cars - YMMV). If budget allows, the CTH and LCX can co-exist nicely in a collection. They are different enough experiences and cater to slightly different use cases. The CTH is a fun, easy to listen too amp. I find myself just enjoying the music on the CTH vs analyzing the music on the LCX. The CTH adds that tubey goodness that seems to just work with the Sennheiser HD line. The LCX resolves the music better, has less roll off on the highs, more extension on the lows which sometimes make it sound like a darker amp depending on the source material - bass goes deep. This can sometimes cause my 6xx's to bloom on the low end (especially with modern recordings with muddier tracks and darker voices like Ray LaMontagne's new album). It also causes the 6xx to lose a little resolution up high causing cymbals and the upper registers of guitar and voice to crash. But with older (read: less dynamic) recordings, the LCX brings out parts of the music I had never noticed before. So if your music collection is all over the map like mine and you have some older stuff that you want to experience differently (I'd say better), then get the LCX. Just don't expect a huge difference...it's all nuances. During my listening session, I went back n' forth between the two amps preferring some tracks on one vs the other...usually based on either the bass blooming or treble crashing some. I found the 6xx with sultry female voices to be a treat on the LCX (Melissa Etheridge Tracy Chapman, Tracey Thorn aka Everything but the Girl). Whereas Elle King and Florence + the Machine needed the smoothness of the CTH. Of interesting note, I'd list Hannah Reid from London Grammar as a sultry voice, but her range is off the hook and certain tracks could go either way on these two amps.
A word on the use of balanced cables... For the purposes of this listening session I based all of my thoughts on using the SE ouputs only. Once I switched to balanced on the LCX, things went (slightly) next level. The already resolving LCX became more so. There was more of everything, but in particular there was more air or expanse (The gated reverb on Melissa Etheridge's voice on the Brave and Crazy album was well...crazy good). Of course with more of everything, there was potential for good and bad. While mostly good, the balanced output also punctuated any flaws in the recordings or in the limitations of the headphones. I think this was why I initially preferred my other cans over the Sennheiser 6xx on the LCX. During that extended session I used the balanced input exclusively. But once I sorted source material, the music I like on the LCX was even better via the balanced output. Side note: I found on songs that could get overly bloomy in the lower registers on the HD6xx sounded nicer on the HD600 with the balance cable. The 600's are well known to be less dark than the 650. Interestingly the HD600's also reduced the crash in the upper register...weird.
Summary... If I had to boil this down to a couple sentences (and yes, I can be wordy)...I'd say this. The CTH/MD Sennheiser HD6xx is a tough combo to beat. Regardless of headphone choice, the CTH is smoother (but not necessarily warmer) easy to listen to and fun. The LCD is more resolving with a larger and more complex sound stage (but not necessarily brighter - in fact can be darker), but certainly more analytical and nuanced. They can easily co-exist, especially if you like to A-B amps, cans and songs. If budget allows, I'd recommend owning both. Even more so if you plan to own super dynamic cans like the Focal Clears, sensitive IEM's or magnetic planar headphones from Mr Speakers, HifiMan or Audeze. Also it's worth owning a set of balanced cables with the LCX.
Hope I'm not too late on this follow-up and hope this proves helpful.