The system variable that you require for this task is AUTOSNAP.
This system variable is bit-coded, with the following bit codes exhibiting the associated behaviour as described below:
0 Turns off the AutoSnap marker, tooltips, and magnet. Also turns off polar tracking, object snap tracking, and tooltips for polar tracking, object snap tracking, and Ortho mode
1 Turns on the AutoSnap marker
2 Turns on the AutoSnap tooltips
4 Turns on the AutoSnap magnet
8 Turns on polar tracking
16 Turns on object snap tracking
32 Turns on tooltips for polar tracking, object snap tracking, and Ortho mode
As such, to control Polar tracking, you'll need to flip bit 8.
Since we're working with a bit-coded integer, it is easiest & most appropriate to manipulate this value using the various bitwise functions defined in AutoLISP: boole, logand, logior, ~. Below are a few examples demonstrating how to accomplish this:
Turn on Polar Tracking:
(setvar 'autosnap (logior 8 (getvar 'autosnap)))
Note that the above is still applicable even when polar tracking is already turned on, since logior implements inclusive OR logic (hence the name "ior"):
_$ (logior 8 39)
47
_$ (logior 8 47)
47
Check if Polar Tracking is enabled:
_$ (= 8 (logand 8 (getvar 'autosnap)))
T
Turn off Polar Tracking:
(setvar 'autosnap (logand (~ 8) (getvar 'autosnap)))
Note that the above is performing a bitwise AND against the 1's-complement of 8 which is 31-bits set to 1 with bit 8 set to 0, that is, returning all bits except bit 8:
_$ (int->bin (~ 8))
"11111111111111111111111111110111"
11111111111111111111111111110111 = (~ 8) = -9
00000000000000000000000000101111 = 47
---------------------------------- LOGAND
00000000000000000000000000100111 = 39
Check if Polar Tracking is disabled:
_$ (zerop (logand 8 (getvar 'autosnap)))
T
Toggle Polar Tracking:
_$ (setvar 'autosnap (boole 6 8 (getvar 'autosnap)))
47
_$ (setvar 'autosnap (boole 6 8 (getvar 'autosnap)))
39
Here, supplying the first argument of 6 to the boole function results in bitwise XOR logic (exclusive OR):
00000000000000000000000000001000 = 8
00000000000000000000000000101111 = 47
---------------------------------- XOR
00000000000000000000000000100111 = 39
00000000000000000000000000001000 = 8
00000000000000000000000000100111 = 39
---------------------------------- XOR
00000000000000000000000000101111 = 47