Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kähler-Weyl structure; this structure is locally conformally Kähler if and only if the alternating Ricci tensor ρa vanishes. The tensor ρa takes values in a certain representation space. In this paper, we show that any algebraic possibility Ξ in this representation space can in fact be geometrically realized by a left-invariant Kähler-Weyl structure on a 4-dimensional Lie group in either the pseudo-Hermitian or the para-Hermitian setting. MSC 2010: 53A15, 53C15, 15A72.