XML-RPC failing to respond to POST requests via cURL in PHP
Managed to fix it. Looking at the headers sent by cURL, the only differences were that the cURL command line uses Content-Type: application/x-www-form-urlencoded and that the user agent was set to User-Agent: curl/7.30.0.
The choice of content type didn't affect it, but setting a user agent sorted it! It seems WordPress.com (but not self-hosted WordPress.org sites running the latest v3.9.2) now requires a user agent for XML-RPC requests, though this hasn't been documented anywhere that I can find.