YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
f(0(),0()) -> f(0(),1())
f(1(),0()) -> f(0(),0())
f(x,y) -> f(y,x)

CP(R):
f(0(),1()) -> f(0(),0())
f(0(),0()) -> f(0(),1())
f(0(),0()) -> f(0(),1())
f(0(),1()) -> f(0(),0())

critical-pair-closing system C:
f(0(),0()) -> f(0(),1())
f(x,y) -> f(y,x)

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
f(0(),0()) -> f(0(),1())
f(x,y) -> f(y,x)
CP(R):
f(0(),1()) -> f(0(),0())
f(0(),0()) -> f(0(),1())