  In a quantum world, reference frames are ultimately quantum systems too --
but what does it mean to "jump into the perspective of a quantum particle"? In
this work, we show that quantum reference frame (QRF) transformations appear
naturally as symmetries of simple physical systems. This allows us to rederive
and generalize known QRF transformations within an alternative, operationally
transparent framework, and to shed new light on their structure and
interpretation. We give an explicit description of the observables that are
measurable by agents constrained by such quantum symmetries, and apply our
results to a puzzle known as the `paradox of the third particle'. We argue that
it can be reduced to the question of how to relationally embed fewer into more
particles, and give a thorough physical and algebraic analysis of this
question. This leads us to a generalization of the partial trace (`relational
trace') which arguably resolves the paradox, and it uncovers important
structures of constraint quantization within a simple quantum information
setting, such as relational observables which are key in this resolution. While
we restrict our attention to finite Abelian groups for transparency and
mathematical rigor, the intuitive physical appeal of our results makes us
expect that they remain valid in more general situations.
