Upside-Down Keypad PIN

A keypad is a $3$ by $3$ grid with the digits $1$ to $3$ in order on the top row, $4$ to $6$ in order on the second row and $7$ to $9$ in order on the third row.

Seb has the keypad upside-down (i.e. rotated through $180^\circ$) and types his three-digit PIN as if the keypad were the right way up.

The number he types in turns out to be $k$ times his original PIN where $k$ is a positive integer.

Find all possible PINs and prove there are no others.