ZKC Controller reported not being able to let a Medevac aircraft go direct to an airport due to D01 Approach control not letting them. Reportedly; this is standard procedure and reporter expressed concern about the confusion it creates.

Narrative

Aircraft X; a MEDEVAC; was filed into ZZZ. He requested priority handling and direct ZZZ with the controller at R24 in ZKC. When coordinated with the controller at ZDV; they were told unable. When asked why; ZDV controller informed ZKC controller that Denver approach does not accept any aircraft; including MEDEVAC; direct to their landing airport. R24 controller informed me; the OS (Operations Supervisor). I called ZDV Sector 17/18 OS to discuss further. ZDV OS said that it was a battle they had fought with Denver approach for years; and gave me the phone number to approach. I called and talked to the OS at Denver approach who said the Denver region had ruled that they were in the right; that they did not have to allow MEDEVAC aircraft to go direct despite their request for priority handling. When asked to elaborate; Approach OS said that it put them in too close proximity to outbound aircraft. I reminded them that when a MEDEVAC requests priority they are authorized to go direct that it is our responsibility as controllers to move other aircraft as necessary to miss them. Frustrated; Approach OS told me I could take it up with their OM (Manager). I then called to talk with their OM. He reiterated the same that the OS had; that they felt it put aircraft in unsafe proximity. I informed him also that it was our responsibility to move aircraft as needed to not delay a MEDEVAC that had requested priority handling. He informed me that the Denver regional office had told them what they were doing was OK. By doing this; they have created a culture that does not accommodate to MEDEVAC aircraft as we are required to do; causing confusion with not only other controllers; but also with pilots of these aircraft that are able to receive the correct priority handling anywhere else they request it. This confusion and refusal to accommodate results in an unsafe situation for the pilots expecting to be handled there the same way they are throughout the rest of the NAS.Denver approach should be appropriately accommodating MEDEVAC aircraft as we are required to do; and not causing confusion for controllers and pilots alike that do not understand why one airport has an inappropriate procedure in place that contradicts the procedures required and adhered to by the rest of the NAS.