Constructing new copulas by transforming a given copula is necessary to enhance the flexibility of the existing copula family. Among various copula transformations, the Khoudraji transformation (K-transformation), described in , provides a promising way to create asymmetric copulas. In this study, we completely characterize the invariance property of the K-transformation and discuss how the K-transformation changes measures of association or asymmetry of a copula. In particular, this study explores the identifiability issue of Khoudraji-transformed copula families. Some conditions for guaranteeing identifiable transformed copula families are provided. (C) 2019 Elsevier B.V. All rights reserved.