We characterize absolutely continuous symmetric copulas with square integrable densities in this paper. This characterization is used to create new copula families, that are perturbations of the independence copula. The full study of mixing properties of Markov chains generated by these copula families is conducted. An extension that includes the Farlie-Gumbel-Morgenstern family of copulas is proposed. We propose some examples of copulas that generate non-mixing Markov chains, but whose convex combinations generate $ψ$-mixing Markov chains. Some general results on $ψ$-mixing are given. The Spearman's correlation $ρ_S$ and Kendall's $τ$ are provided for the created copula families. Some general remarks are provided for $ρ_S$ and $τ$. A central limit theorem is provided for parameter estimators in one example. A simulation study is conducted to support derived asymptotic distributions for some examples.