I won't bother you with all the boring details but suffice it to say that I ended up with two sites with completely identical theme folders and the Anomaly theme (didn't try others) working in one but not in the other.
The difference was a couple of custom blocks that we had sub-contracted out.
It turned out that the dev had obviously used some horrid Windows editor which had added byte-order-makrs (BOMs) to all the block files. This was enough to chuck IE into Quirks Mode and mash the theme. The only requirement was to be logged in, not to have the block displayed (it got even worse with the block displayed).
So... the curse of Microsoft strikes again!!
PS: If you want to know how to check for BOMs in files:
grep -rl $'\xEF\xBB\xBF' *
(which only works on real operating systems.... of course!!)