  Starting from the original Einstein action, sometimes called the Gamma
squared action, we propose a new setup to formulate modified theories of
gravity. This can yield a theory with second order field equations similar to
those found in other popular modified gravity models. Using a more general
setting the theory gives fourth order equations. This model is based on the
metric alone and does not require more general geometries. It is possible to
show that our new theory and the recently proposed $f(Q)$ gravity models are
equivalent at the level of the action and at the level of the field equations,
provided that appropriate boundary terms are taken into account. Our theory can
also match up with $f(R)$ gravity which is an expected result. Perhaps more
surprisingly, we can also show that this equivalence extends to $f(T)$ gravity
at the level of the action and its field equations, provided that appropriate
boundary terms are taken in account. While these three theories are
conceptually different and are based on different geometrical settings, we can
establish the necessary conditions under which their field equations are
indistinguishable. The final part requires matter to couple minimally to
gravity. Through this work we emphasise the importance played by boundary terms
which are at the heart of our approach.
