First, we set a suitable notation. Points in {0, 1} Z−{0} = {0, 1} N × {0, 1} N = Ω − × Ω + , are denoted by (y|x) = (..., y 2 , y 1 |x 1 , x 2 , ...), where (x 1 , x 2 , ...) ∈ {0, 1} N , and (y 1 , y 2 , ...) ∈ {0, 1} N . The bijective map σ(..., y 2 , y 1 |x 1 , x 2 , ...) = (..., y 2 , y 1 , x 1 |x 2 , ...) is called the bilateral shift and acts on {0, 1} Z−{0} . Given A : {0, 1} N = Ω + → R we express A in the variable x, like A(x). In a similar way, given B : {0, 1} N = Ω − → R we express B in the variable y, like B(y). Finally, given W : Ω − × Ω + → R, we express W in the variable (y|x), like W (y|x). By abuse of notation we write A(y|x) = A(x) and B(y|x) = B(y). The probability µ A denotes the equilibrium probability for A : {0, 1} N → R.Given a continuous potential A : Ω + → R, we say that the continuous potential A * : Ω − → R is the dual potential of A, if there exists a continuous W : Ω − × Ω + → R, such that, for all (y|x) ∈ {0,We say that W is an involution kernel for A. The function W allows you to define an spectral projection in the linear space of the main eigenfunction of the Ruelle operator for A. Denote by θ : Ω − × Ω + → Ω − × Ω + the function θ(..., y 2 , y 1 |x 1 , x 2 , ...) = (..., x 2 , x 1 |y 1 , y 2 , ...). We say that A is symmetric if A * (θ(x|y)) = A(y|x) = A(x). To say that A is symmetric is equivalent to saying that µ A has zero entropy production. Given A, we describe explicit expressions for W and the dual potential A * , for A in a family of functions introduced by P. Walters. We present conditions for A to be symmetric and to be of twist type.