A321 Captain reported a fumes event described as a strong dirty socks odor resulting in health issues. The odor was first noticed during boarding; but it dissipated after the engines were started. The odor returned during climbout at which point the crew decided to divert.

Narrative

At approximately XA00L; the First Officer and I boarded Aircraft X in ZZZZ. The First Officer immediately commented on the very-heavy dirty-sock odor in the cockpit; I agreed; and confirmed that the cabin was significantly affected as well. I asked the inflight crew to give their assessment; and they all agreed that we should not board under these conditions. I advised local maintenance about the issue. Maintenance confirmed the presence of the odor; and was able to clear the odor without an engine run and by deferring the APU Bleed. I made the appropriate logbook entry and completed the Odor Event form in the logbook.After starting both engines; the odor was not present. We departed Gate X in ZZZZ and were climbing through approximately 2;000 ft. MSL when the odor returned; though not to the same degree as at the gate in ZZZZ. We had climbed through an extremely thin layer of haze with extremely light precipitation. The First Officer commented that the odor was affecting him considerably; so as PM (Pilot Monitoring) I advised ATC of the odor event and requested a PPOS hold to assess the situation. Instead; ATC gave us a heading and advised that we would be vectored.The First Officer stated that he was not able to continue with the odor; as it was affecting him physically. Because we were overweight; I decided on a diversion to ZZZZ1; and contacted Dispatch via SATCOM to confirm my intention. After Dispatch concurred; I advised inflight about the diversion with a Code Yellow; and made an announcement to the customers. ATC cleared us to ZZZZ1 and asked if we were [requesting priority]. I replied that we were not; but ATC said they were [giving us priority handling] on our behalf; if I remember correctly; to 'get us into ZZZZ1.'We referenced the FOM and QRH procedures for an Odor Event and Overweight Landing; respectively. We set up for the longer [Runway] XX at ZZZZ1. I assigned myself as PF (Pilot Flying) to continue the flight. I landed at ZZZZ1 with a V/S of less than 100FPM. We were met at the gate by CFR (Crash Fire Rescue); and confirmed that there was no smoke present per CFR inquiry. I did not understand why CFR was asking me this question; but then realized that I may have stated an odor or fumes event to ATC; which would explain the [priority handling] on our behalf.We were also met by local maintenance; to whom I provided a detailed account of the Odor Event beginning in ZZZZ and the Overweight Landing in ZZZZ1. I made the appropriate logbook entries for the Odor Event and Overweight Landing; and completed another Odor Event form in the logbook.The First Officer contacted [Medical Services]. Paramedics arrived at the aircraft to evaluate the First Officer; who was removed from the pairing.I had several conversations with the Duty Chief Pilot about the event; and was initially told we would be overnighting in ZZZZ1. However; the entire crew; except the First Officer; was reassigned to operate from ZZZZ1 to ZZZ. I confirmed that all crewmembers were fit to fly; and we operated Aircraft Y on a different aircraft to ZZZ.