This paper provides a classification theorem for expansive matrices A ∈ GL(d, R) generating the same anisotropic homogeneous Triebel-Lizorkin space Ḟα p,q (A) for α ∈ R and p, q ∈ (0, ∞]. It is shown that Ḟα p,q (A) = Ḟα p,q (B) if and only if the homogeneous quasi-norms ρA, ρB associated to the matrices A, B are equivalent, except for the case Ḟ0p,2 = L p with p ∈ (1, ∞). The obtained results complement and extend the classification of anisotropic Hardy spaces H p (A) = Ḟ0 p,2 (A), p ∈ (0, 1], in [Mem. Am. Math. Soc. 781, 122 p. (2003)].