Inverting polarity will only help if the snare was out of phase to begin with. More commonly, check the bottom snare signal polarity against the top snare. But if you are using programmed drums, that probably won't be an issue.
Sounds more like you're neglecting the lower mids, overcompressing the snare, and not using a snare reverb. Those all contribute to "papery" sound.
Programmed drums usually come in "eq'd". Over extending the upper mids might sound good - but really it might just sound better to turn the snare up. Similarly, don't neglect that the 200-400hz area of a snare is the beef and depth of it. There's often times some tones around 500hz that I don't like in an acoustic snare, and sometimes another tone around 800hz that kind of sounds like a "pong" noise. But below that and above that there tends to be a lot of good stuff. This is all generality mind you.
Snare compression is a little tricky. I tend to do it it two stages. The first is some kind of dynamic processing to enhance the attack if I'm not already getting a good punch. You can use a compressor like an 1176 with a slow attack, or a gate, to bring out the attack signal. The second step for me is hard compression or limiting. So after I exaggerate the attack, I then give it a little squish. I know that sounds counterintuitive, but it's a good technique for getting a very defined and thick attack sound. But often you don't really need all that. Don't underestimate the power of simply setting level.
An alternative to compression techniques is using mild distortion. This has a similar way of thickening the attack. You can do this using a tape machine, tape emulator, digidesign lo-fi, decapitator, hitting the board channels hot... there's a million ways. You get a slight bit of break up on the peak of the snare signal which excites it and makes the attack a bit more potent.
Lastly, acoustic snares and reverbs have been dating a long time. They're sort of meant to be. A short plate or spring reverb, mono, tucked under the snare gives it lift and dimension. You don't really need to hear it much - just feel it as "part of the snare." I like to set the reverb so the low end doesn't damp out too fast. This helps give the snare weight. Unless it's already a weighty snare.