I understand that an id must be unique within an HTML/XHTML page.
My question is, for a given element, can I assign multiple ids to it?
No. While the definition from w3c for HTML 4 doesn't seem to explicitly cover your question, the definition of the name and id attribute says no spaces in the identifier:
ID and NAME tokens must begin with a letter ([A-Za-z]) and may be followed by any number of letters, digits ([0-9]), hyphens ("-"), underscores ("_"), colons (":"), and periods (".").