The word right has many meanings, among them a few verb senses—namely, (1) to set upright, (2) to put in order, (3) to correct, and (4) to make reparations or amends. Because right functions as a verb, righten is superfluous.
For example, righten would bear replacement with right in these sentences:
Alter (1-2) is hoping to righten its way at the St. Pius X Catholic tournament in Atlanta. [Dayton Daily News]
Righten the ivory tower with free speech power [Post and Courier]
And in these sentences, right is a verb:
But Walker was somehow dexterous enough to right himself and float in the game-winner. [Wall Street Journal]
In our own little company we were occupied with a lawsuit and righting wrongs against the “Odious Bruja” and “Mr Slick”. [Financial Times]
Reagan had righted the sinking ship by raising taxes a record amount. [Los Angeles Times]