Robot factory permutation riddle

[mathnice1]

A factory has built a robot which moves on all squares of a 6*6 table. There is an arrow On all of the squares of the table when it moves on an square it reads the arrow the square and moves according to the square but before doing that it changes the arrow of the square aim to the previous square(except the 1st square because there was not any squares before that .)...after some moving we understand that the robot has moved on all of the squares for 2 orders (except the first square) show that all of the arrows are the same as the first
sorry for my bad English

 

[HallsofIvy]

I assume that "moves according to the square" means that it moves to the square the arrow is pointing to. Does "has moved on all of the squares for 2 orders" mean it has moved twice over each square?

 

[mathnice1]

I assume that "moves according to the square" means that it moves to the square the arrow is pointing to. Does "has moved on all of the squares for 2 orders" mean it has moved twice over each square?

exactly

 

[marcusmath]

Surely if the robot has passed over each square exactly twice, the robot has followed some path from the first square to the last, and the last square he stands on changes to point back to the square he came from.
This means he will follow that same path to get back to the start and will have stepped on each square twice, thereby reverting all the arrows to their original direction.

EDIT:I denoted the robot as a 'he', I apologize if the robot is actually a female :)