Let \(X\) be a finite-dimensional Banach space; we introduce and investigate a natural generalization of the concepts of Hadwiger number \(H(X)\) and strict Hadwiger number \(H'(X)\). More precisely, we define the antipodal Hadwiger number \(H_\alpha(X)\) as the largest cardinality of a subset \(S \subseteq S_X\), such that \(\forall x \neq y \in S \,\,\, \exists f \in B_{X^*}\) with \1 \le f(x)-f(y) \,\,\, \textrm{and} \,\,\, f(y) \le f(z) \le f(x) \,\,\, \textrm{for} \,\,\, z \in S.\ The strict antipodal Hadwiger number \(H'_\alpha(X)\) is defined analogously. We prove that \(H'_\alpha(X)=4\) for every Minkowski plane and estimate (or in some cases compute) the numbers \(H_\alpha(X)\) and \(H'_\alpha(X)\), where \(X=\ell_p^n, 1 < p \le +\infty\) and \(n \ge 2\). We also show that the number \(H'_\alpha(X)\) grows exponentially in \(\dim X\).