I saw a Facebook promotion for a video with the words “sends a shiver up your spine”. I have only ever seen or used “a shiver down your spine”. Are both used?

Yes, as well as "sends a shiver up AND down my spine"

IIRC, shiver is related to shard and to shiv, a knife. Who wants shrapnel down the spine?

It must be a U.S. way of using it because I've heard sends a shiver up your spine too.

In Eric Larson's biography of Churchill, The Splendid and The Vile, a Londoner, referring to Churchill during the London Blitz, said, "Winston's speeches send all sorts of thrills racing up and down my veins..."

Now, for some idiotic reason, I wonder if taxidermy makes your skin crawl.

Interesting, Geoff. I'd surely say "racing up my veins..."