  A PCSP is a combination of two CSPs defined by two similar templates; the
computational question is to distinguish a YES instance of the first one from a
NO instance of the second. The computational complexity of many PCSPs remains
unknown. Even the case of Boolean templates (solved for CSP by Schaefer
[STOC'78]) remains wide open.
  The main result of Brakensiek and Guruswami [SODA'18] shows that Boolean
PCSPs exhibit a dichotomy (PTIME vs. NPC) when "all the clauses are symmetric
and allow for negation of variables''. In this paper we remove the "allow for
negation of variables'' assumption from the theorem. The "symmetric" assumption
means that changing the order of variables in a constraint does not change its
satisfiability. The "negation of variables" means that both of the templates
share a relation which can be used to effectively negate Boolean variables.
  The main result of this paper establishes dichotomy for all the symmetric
boolean templates. The tractability case of our theorem and the theorem of
Brakensiek and Guruswami are almost identical. The main difference, and the
main contribution of this work, is the new reason for hardness and the
reasoning proving the split.
