A320 Captain reported a fumes event during initial climb. After conferring with Dispatch the flight diverted to a suitable alternate.

Narrative

All aircraft systems were operating normally; APU used for engine start. No de-icing of aircraft; engine anti-ice selected on for takeoff and engine bleeds on. During climb to cruise at 10;000 ft.; Inflight called and notified that the crew and passengers were smelling 'dirty sock' odor in the cabin. We also began to smell the odor in the cockpit. Cockpit / Cabin Odor Recognition and Isolation; was referenced and flight deck oxygen masks were deployed. Inflight was called and advised to utilize oxygen as well if necessary. The odor was reported as increasing and intensifying in the cabin. Per the guidelines; the intensity was determined to be [low] initially but increased. A message was sent to Operations advising of the situation and requesting diversion options. Operations indicated ZZZ as the preferred divert destination and we concurred. ATC was notified and we received clearance to ZZZ. Dispatch sent planning information. A passenger PA was made explaining the cabin air issue and that we would be returning to land at ZZZ. ATC provided expedited handling of our arrival and thus no emergency was declared. Inflight was advised of information. An approach; landing and gate arrival were completed without incident. PACs 1 and 2 were deselected when on the ground. None of the passengers or crew requested or showed indication of needing immediate medical assistance.Maintenance met the aircraft; a logbook entry was made and a cabin odor report was completed.