Python 3, 98 74 bytes

Saved a whopping 24 bytes thanks to ovs!!!

import re
f=lambda p:p[1:]and f(re.sub('\(((.)<.|.>(.))\)',r'\2\3',p))or p

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Python 3, 98 74 bytes

Saved a whopping 24 bytes thanks to ovs!!!

import re
f=lambda p:p[1:]and f(re.sub('\(((.)<.|.>(.))\)',r'\2\3',p))or p

Try it online!

Takes an input string with enclosing in parentheses and returns whether it's left (<) or right (>).

Python 3, 98 bytes

import re
f=lambda p:p*(len(p)<2)or f(re.sub(r'\((.)<(.)\)',r'\1',re.sub(r'\((.)>(.)\)',r'\2',p)))

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Python 3, 98 74 bytes

Saved a whopping 24 bytes thanks to ovs!!!

import re
f=lambda p:p[1:]and f(re.sub('\(((.)<.|.>(.))\)',r'\2\3',p))or p

Try it online!