Pin-change is a digital function. It is only interested in the digital level HIGH or LOW. It can't see that the analog input which was LOW is now LOWER.
If it's just a wakeup, then a quick waggle left-right is probably OK to wake the machine up. If you actually want to read 'left' without first going right or pressing another button, then you have work to do.
A comparator is able to compare two analog voltages and change a digital output as a result. Most (all?) Arduino chips do have an analog comparator module. It is rarely used and there aren't any simple Arduino functions to make it easy to use. The comparator can be used to generate an interrupt. But there's only one comparator. I don't even know if it could be applied to the X and Y inputs at the same time.
You could also add an external comparator. Something that will give a change on a digital pin when the joystick goes left or down. I'm thinking that it might be possible to add a couple of resistors to tap off that signal and push it to the other side of the high/low decision point, so that a LOW to LOWER transition pulls another pin from HIGH to LOW.