I learned I had HPV after finding anal warts and going to the doctor to check them out. The other thing about there not being a test for the virus to confirm is which strain of the virus you might have. I likely have one of two strains related to anal warts and the vaccine only covers a single or certain strains, not all. I got the vaccine after the diagnosis for the future and with any chance to reduce the warts or fight the virus at all, although they say the virus could never go away. My warts did go away not long after the third/final round but that may just be coincidence based on the lifespan of the warts.
Someone mentioned an increase risk in rectal cancer in men from HPV, where did you see this? I don't remember seeing such a risk in my searches. I was under the impression the virus had non-cancer effects in men.
Another point about this virus I feel isn't widely known is you can still get it with protected sex. It can be transmitted by skin-on-skin contact. That is, the skin around the crotch, outside of a condom, against a rear end for example in anal sex. Scary thought in my mind.
In regards to after effects from the vaccine, I didn't notice or remember anything other than the typical sore arm as it the same type of shot as the flu.
If you really want to know what strain it is, your doctor can have PCR test ran. But its probably not going to be cheep and why he didn't mention it. I'd say its still worth asking about it so you can both have a better understanding of your long term treatment.