  We focus on a paradigmatic two-dimensional model of a nanoscale heat engine,
- the so-called Brownian gyrator - whose stochastic dynamics is described by a
pair of coupled Langevin equations with different temperature noise terms. This
model is known to produce a curl-carrying non-equilibrium steady-state with
persistent angular rotations. We generalize the original model introducing
constant forces doing work on the gyrator, for which we derive exact asymmetry
relations, that are reminiscent of the standard fluctuation relations. Unlike
the latter, our relations concern instantaneous and not time averaged values of
the observables of interest. We investigate the full two-dimensional dynamics
as well as the dynamics projected on the $x$- and $y$-axes, so that information
about the state of the system can be obtained from just a part of its degrees
of freedom. Such a state is characterized by effective "temperatures" that can
be measured in nanoscale devices, but do not have a thermodynamic nature.
Remarkably, the effective temperatures appearing in full dynamics are
distinctly different from the ones emerging in its projections, confirming that
they are not thermodynamic quantities, although they precisely characterize the
state of the system.
