Forman et al. (2020+) constructed $(alpha,theta)$-interval partition evolutions for $alphain(0,1)$ and $thetage 0$, in which the total sums of interval lengths (total mass) evolve as squared Bessel processes of dimension $2theta$, where $thetage 0$ acts as an immigration parameter. These evolutions have pseudo-stationary distributions related to regenerative Poisson--Dirichlet interval partitions. In this paper we study symmetry properties of $(alpha,theta)$-interval partition evolutions. Furthermore, we introduce a three-parameter family ${rm SSIP}^{(alpha)}(theta_1,theta_2)$ of self-similar interval partition evolutions that have separate left and right immigration parameters $theta_1ge 0$ and $theta_2ge 0$. They also have squared Bessel total mass processes of dimension $2theta$, where $theta=theta_1+theta_2-alphage-alpha$ covers emigration as well as immigration. Under the constraint $max{theta_1,theta_2}gealpha$, we prove that an ${rm SSIP}^{(alpha)}(theta_1,theta_2)$-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson--Dirichlet distribution with parameters $alpha$ and $theta$, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper.
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