Of course, the anesthesia provider may report medically necessary surgical procedures, however, E&M services are not allowed unless it would qualify as critical care (which then, per critical care billing instructions, total critical care time would exclude any procedure time).
The below excerpt is from the NCCI Chapter 2 (Anesthesia) page 5
"Similarly, routine postoperative evaluation is included in the basic unit for the anesthesia service. If this evaluation occurs after the anesthesia practitioner has safely placed the patient under postoperative care, neither additional anesthesia time units nor evaluation and management codes should be reported for this evaluation. Postoperative evaluation and management services related to the surgery are not separately reportable by the anesthesia practitioner except when an anesthesiologist provides significant, separately identifiable ongoing critical care services.
Anesthesia practitioners other than anesthesiologists cannot report evaluation and management codes except as described above when a surgical case is canceled."
Julie, CPC