I've got a few things to add regarding control after gaining more experience over the years.
At a beginner level, I think hardbat offers a player more control because it is more linear. Inverted has more levels/dimensions though, and I think as you progress and play stronger players, inverted offers more control for attacking strokes (or even topspin defending) and defensive long pips offer more control for defensive strokes.
You get more topspin with inverted, which creates more of an arc over the net leading to safer attacking shots. You can go for more with less fear of missing the table. Hardbat, in terms of attacking, only really excels at flat hitting when the ball is higher than the net. It's amazingly unforgiving though if your technique is poor or if you're not in the right position. The higher in level you go though, the less 'ideal' chances you will have to attack with a hardbat vs inverted.
Defensively, a hardbat is amazing against players who don't possess much power or spin. You can vary how much chop is on the ball easily with just slight movements (which will lead players who don't understand pips to accuse you of being a bat and not a player
). However, against strong spin and pace, defending with a hardbat requires everything to be just right. Your footwork has to be right, your technique has to be solid, and you have to be far enough from the table to take the ball as low as possible (similar to chopping with a high speed inverted). If you get caught chopping the ball above your waist, you'll have practically no control over the chop (if there is a lot of pace/spin on the ball). With enough experience and technique, this is obviously doable, but a strong enough inverted attacker will be able to keep up a steady onslaught. The more chops you have to do with a hardbat against this kind of player, the higher the chance you will get caught in a bad position.
This is where defensive long pips come in, and it's only this year that I've started to get why. I think this is why you don't see many SP defenders on the pro tour as well. A quality defensive long pip will give you control over most chops against high spin and pace, even if you're caught close to the table. Long pips are assisted defending, just like fast inverted rubbers offer assisted attacking.
A hardbat doesn't do a lot for you at higher levels I don't think (obviously against inverted). I think at this stage playing with a hardbat against inverted is likely more of a stance (such as 'I love hardbat'). Hardbat vs hardbat is great fun. Hardbat vs inverted is more like putting the difficulty up to hard in a video game lol.
I will say though that having SP OX on one side and inverted on the other could be a very effective combination. It's when you have it on both sides (specifically if it's the same SP OX on both sides), which is what I refer to when I talk about hardbat, that the problems mount up as you go up in level.