I was covering for another physician and saw one of his established patients. Do I bill for a new-patient visit?

No. When you're covering for another physician's patient, and don't assume care, it's considered an established-patient visit. Medicare explicitly recognizes cross coverage (referred to as reciprocal coverage) arrangements and allows either the covered or the covering physician to bill, with different modifiers in each instance. These modifiers explicitly notify the carrier of the coverage arrangement. The arrangement is fine as long as the covered physician is treating the patient and the covering physician does not provide the services for longer than 60 continuous days.