The purpose of this work is to classify, for given integers $${m,\, n\geq 1}$$ m , n ≥ 1 , the bordism class of a closed smooth $${m}$$ m -manifold $${X^m}$$ X m with a free smooth involution $${\tau}$$ τ with respect to the validity of the Borsuk–Ulam property that for every continuous map $${\phi : X^m \to \mathbb{R}^n}$$ φ : X m → R n there exists a point $${x\in X^m}$$ x ∈ X m such that $${\phi (x)=\phi (\tau (x))}$$ φ ( x ) = φ ( τ ( x ) ) . We will classify a given free $${\mathbb{Z}_2}$$ Z 2 -bordism class $${\alpha}$$ α according to the three possible cases that (a) all representatives $${(X^m, \tau)}$$ ( X m , τ ) of $${\alpha}$$ α satisfy the Borsuk–Ulam property; (b) there are representatives $${({X_{1}^{m}}, \tau_1)}$$ ( X 1 m , τ 1 ) and $${({X_{2}^{m}}, \tau_2)}$$ ( X 2 m , τ 2 ) of $${\alpha}$$ α such that $${({X_{1}^{m}}, \tau_1)}$$ ( X 1 m , τ 1 ) satisfies the Borsuk–Ulam property but $${({X_{2}^{m}}, \tau_2)}$$ ( X 2 m , τ 2 ) does not; (c) no representative $${(X^m, \tau)}$$ ( X m , τ ) of $${\alpha}$$ α satisfies the Borsuk–Ulam property.