  This paper is devoted to studying $R$-diagonal and $\eta$-diagonal pairs of
random variables. We generalize circular elements to the bi-free setting,
defining bi-circular element pairs of random variables, which provide examples
of $R$-diagonal pairs of random variables. Formulae are given for calculating
the distributions of the product pairs of two $*$-bi-free $R$-diagonal pairs.
When focusing on pairs of left acting operators and right acting operators from
finite von Neumann algebras in the standard form, we characterize $R$-diagonal
pairs in terms of the $*$-moments of the random variables, and of
distributional invariance of the random variables under multiplication by free
unitaries.
  We define $\eta$-diagonal pairs of random variables, and give a
characterization of $\eta$-diagonal pairs in terms of the $*$-distributions of
the 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 its
adjoint $x^*$ in the algebra and the diagonal $2\times 2$ scalar matrices can
never be Boolean independent fromm the $2\times 2$ scalar matrix algebra with
amalgamation over the diagonal scalar matrix algebra.
