Yes, of course he can as long has he stays within 3 feet of a line directly between him and either base, he can go either direction. He's still obligated to get to the base ahead of him eventually if there's a force.
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MLB Rule 6.03 states: "The batter's legal position shall be with both feet within the batter's box. "
MLB Rule 6.06(a) states: A batter is out for illegal action when --
(a) He hits a ball with one or both feet on the ground entirely outside the batter's box.
While swinging at a pitch, the batter must have part of both feet within the batter's box.
If you mean running in the reverse direction along the baseline as in home-3rd-2nd-1st-home, then here are some possible answers from the 2009 Official Rules of Major League Baseball: "7.08: Any runner is out when -- 7.08 (i) After he has acquired legal possession of a base, he runs the bases in reverse order for the purpose of confusing the defense or making a travesty of the game. The umpire shall immediately call "Time" and declare the runner out; Rule 7.08 (i) Comment: if a runner touches an unoccupied base and then thinks the ball was caught or is decoyed into returning to the base he last touched, he may be put out running back to that base, but if he reaches the previously occupied base safely he cannot be put out while in contact with that base."
It's not wise. A batter running towards the playing field can be mistaken for trying to take second base.
The manager won't let him do it anyway. They are drilled from little league on up never to turn in towards the playing field. By the time they are in the pros, it's second nature.