If you are using it as a three position switch, you will need more contacts, because the cam will only override it to the same state whether you turn it left or right, I assume you are wanting to turn off when left, on when right, with center auto. You do not have enough contacts to do that. That's why it's effectively a two position switch unless you add more contacts. No biggie to do though. As we both have said, until we find out what the engineer wants, it's all speculation.
Look at it again.
When you turn the key to left "off" position, only the left switch blocks change state. Left N.C. opens, breaks time clock ON to C1 CLOSE. Left N.O. closes, makes L to C1 OPEN. If the time clock status is OFF, then effectively no change in C1 status. If the time clock state is ON, then it has effectively removed time clock ON from cycling C1 status.
When you turn the key to the right position, the right blocks change state. Left N.C. opens, breaks time clock OFF to C1 OPEN. Left N.O. closes, makes L to C1 CLOSE. If the time clock status is ON, then effectively no change in C1 status. If the time clock state is OFF, then the right N.C. block has effectively removed time clock OFF from cycling C1 status.
When the key is returned to the center position, switch blocks return to their normal state. Control is returned to the time clock through both N.C. blocks Whatever state the time clock is in at the time, it will trigger C1 to the matching state, i.e. time cock ON/OFF, C1 CLOSE/OPEN respectively, if C1 is in the opposite state. Don't forget about the clearing contacts in C1.