Much of this adjustability is different with the different AVR manufacturers. I will use Onkyo as an example since I know their features the best. Most Onkyo receivers have an adjustable manual EQ. They also have tone controls that can be adjusted along with the manual EQ. This is the part you need to remember. If you're using Audyssey you cannot use the manual EQ. You can however still use the tone control. Which is bass and treble only. No midrange control. I don't think any modern AVRs have a mid control in the tone control. Just bass and treble. So with an Onkyo you could adjust the mid area with the manual EQ if you weren't using Audyssey.
Yamaha has a pretty good manual EQ, it is a limited parametric EQ but it has a lot more flexibility than most receivers that have a manual graphic EQ. You can also copy the Yamaha YPAO settings from its auto EQ and then tweak from there if you like. Most receivers you have to choose between manual and their auto EQ like Audyssey.