The police CAN order it towed for several reasons (blocking traffic, undrivable, evidence, etc) and the officer has to fill out a form (in California it is called a 180CHP form) which puts the liability on the police department.
I would request a copy of the form from the PD, and file a claim with the PD for the damages. Let them hash it out with the tow company as far as who is going to pay for the damage.
HOWEVER, there's a good chance that the tow is going to be listed as "owners request" because that is common for accident tows with an officer just saying to the driver "you want this towed right?" or something similar(which is a backwards way of obtaining an owners request to tow) because they don't want to deal with the logistics of the damaged car being on the side of the road where it can get hit again.
If the car was not blocking traffic, was driveable, was not needed for evidence and was legally parked then the officer would have had to obtain the owner's request to tow it before telling the driver to put it on the truck. If the officer just assumed the owner wanted it towed and didn't get a verbal or written OK to tow it, then the officer is buying that liability. Odds are the officer did get consent from DH is a roundabout way (and should have documented that conversation in the report) and DH didn't realize it. Or the officer considers the handing over of the keys as consent.
I don't think PD's do this because they are in cahoots with tow companies, the PD gets nothing from the tow company but a guarantee of reliability (your tow company will have 5 trucks available at all times when we call in exchange for us using your service exclusively). It is just easier to get the cars out of there and not have traffic backing up or more accidents because of the parked, wrecked vehicle.