C (gcc), 120 bytes

i;f(char*p){for(i=0;p[i]-40||p[i+4]-41;++i);p[i]=p[i+2]-62?p[i+1]:p[i+3];memcpy(p+i+1,p+i+5,strlen(p));i=*p-40?*p:f(p);}

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