One vertical and one horizontal? That's a 20a circuit alright ... but it could also be 220v as well. By any chance does it look like this one?
http://www.frentzandsons.com/Hardware References/plug imiges/5-20p.jpg
If so, then it's 110v 20a - your plan of action is okay - wire up with 12ga wire and a 20a breaker. It's absolutely a safety concern if you switch that plug out to a 15a plug and run it on "high". If the breaker you're using doesn't trip, fire won't be far behind.
On the other hand... If it looks like this:
http://www.frentzandsons.com/Hardware References/plug imiges/6-20p.jpg
Notice the OTHER prong is sideways. You've got a 220v welder and will need a 220v circuit to run it. 220v welders are pretty common, for sure. It also is rated for 20a, but that means 20a on TWO legs, which is twice as much power as the 110v version. You'd still wire it up with 12ga wire (and 12/2 is fine, just mark the white wire with black to show it's hot), but you'd have to use a 220v breaker, which will take up double the space in your box.
I'm oversimplifying things electrically speaking because we're not 100% sure which way you need to go. Given the specs you have given, I'd almost bet you've got a 220v welder. A 20% duty at 70A sounds like a spec i'd only see on a 220v welder - though i'm far from an expert on all models out there.
It's very important to get this worked out before you go stickin metal together, though. If you have any electrician friends who owe you a favor, now's the time to call 'em!