Well, a pleasant surprise! It _is_ possible to [tile a 2x2x2 cube
with distinct Rubik's Tangle pieces, uniquely]
I'll include the solution below, after some blank lines to avoid
spoiling it for anyone who wants to try solving the puzzle without
seeing the answer...
You may have noticed it yourself, but the solution you promised was
missing from your message.
Not from the copy I got - though the lines in question weren't blank;
they each contained a dot. Perhaps someone's mailer isn't doing the
hidden-dot algorithm correctly?