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This paper is devoted to studying $R$-diagonal and $\eta$-diagonal pairs ofrandom variables. We generalize circular elements to the bi-free setting,defining bi-circular element pairs of random variables, which provide examplesof $R$-diagonal pairs of random variables. Formulae are given for calculatingthe distributions of the product pairs of two $*$-bi-free $R$-diagonal pairs.When focusing on pairs of left acting operators and right acting operators fromfinite von Neumann algebras in the standard form, we characterize $R$-diagonalpairs in terms of the $*$-moments of the random variables, and ofdistributional invariance of the random variables under multiplication by freeunitaries.  We define $\eta$-diagonal pairs of random variables, and give acharacterization of $\eta$-diagonal pairs in terms of the $*$-distributions ofthe random variables.  If every non-zero element in a $*$-probability space has a non-zero$*$-distribution, we prove that the unital algebra generated by a $2\times 2$off-diagonal matrix with entries of a non-zero random variable $x$ and itsadjoint $x^*$ in the algebra and the diagonal $2\times 2$ scalar matrices cannever be Boolean independent fromm the $2\times 2$ scalar matrix algebra withamalgamation over the diagonal scalar matrix algebra.

$R$-diagonal and $\eta$-diagonal Pairs of Random Variables