HTC One X hangs at bootscreen, repair centre writes it off
So I flashed the TWRP recovery instead of ClockworkROM, and this was able to activate mass storage mode. The rest was history - copied the ROM zip file and installed it as usual. 
Sadly, my internal 32 GB storage was totally wiped out, so I've lost my TitaniumBackup folder, a bunch of songs (that I can copy again anyway) and some photos.