What is modified gravity and how to differentiate it from particle dark
  matter?
  An obvious criterion to classify theories of modified gravity is to identify
their gravitational degrees of freedom and their coupling to the metric and the
matter sector. Using this simple idea, we show that any theory which depends on
the curvature invariants is equivalent to general relativity in the presence of
new fields that are gravitationally coupled to the energy-momentum tensor. We
show that they can be shifted into a new energy-momentum tensor. There is no a
priori reason to identify these new fields as gravitational degrees of freedom
or matter fields. This leads to an equivalence between dark matter particles
gravitationally coupled to the standard model fields and modified gravity
theories designed to account for the dark matter phenomenon. Due to this
ambiguity, it is impossible to differentiate experimentally between these
theories and any attempt of doing so should be classified as a mere
interpretation of the same phenomenon.
