RCurl: HTTP Authentication When Site Responds With HTTP 401 Code Without WWW-Authenticate
I've resolved the problem with the help of the author of RCurl, Duncan Lang. The solution is to explicitly set the httpauth option which sets the authentication method to try initially. This works:
getURL("https://api.picloud.com/job/?jids=12", userpwd="key:secret", httpauth = 1L)
httpauth is a bitmask specifying which authentication methods to use. See the HTTP Authentication section of the libcurl tutorial for more details.