  Standard results in descriptive set theory provide sufficient conditions for
a set $P \subseteq \mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N}$ to admit
a Borel uniformization, namely, when $P$ has ''small'' sections or ''large''
sections. We consider an invariant analogue of these results: Given a Borel
equivalence relation $E$ and an $E$-invariant set $P$ with ''small'' or
''large'' sections, does $P$ admit an $E$-invariant Borel uniformization?
  Given $E$, we show that every such $P$ admits an $E$-invariant Borel
uniformization if and only if $E$ is smooth. We also compute the definable
complexity of counterexamples in the case where $E$ is not smooth, using
category, measure, and Ramsey-theoretic methods.
  We provide two new proofs of a dichotomy of Miller classifying the pairs $(E,
P)$ such that $P$ admits an $E$-invariant uniformization, for $P$ with
countable sections. In the process, we prove an $\aleph_0$-dimensional $(G_0,
H_0)$ dichotomy, generalizing dichotomies of Miller and Lecomte. We also show
that the set of pairs $(E, P)$ such that $P$ has ''large'' sections and admits
an $E$-invariant Borel uniformization is $\boldsymbol{\Sigma^1_2}$-complete; in
particular, there is no analog of Miller's dichotomy for $P$ with ''large''
sections.
  Finally, we consider a less strict notion of invariant uniformization, where
we select a countable nonempty subset of each section instead of a single
point.
