  Modified gravity models require a screening mechanism to be able to evade the
stringent constraints from local gravity experiments and, at the same time,
give rise to observable astrophysical and cosmological signatures. Such
screened modified gravity models necessarily have dynamics determined by
complex nonlinear equations that usually need to be solved on a model-by-model
basis to produce predictions. This makes testing them a cumbersome process. In
this paper, we investigate whether there is a common signature for all the
different models that is suitable to testing them on cluster scales. To do this
we propose an observable related to the fifth force, which can be
observationally related to the ratio of dynamical-to-lensing mass of a halo,
and then show that the predictions for this observable can be rescaled to a
near universal form for a large class of modified gravity models. We
demonstrate this using the Hu-Sawicki $f(R)$, the Symmetron, the nDGP, and the
Dilaton models, as well as unifying parametrizations. The universal form is
determined by only three quantities: a strength, a mass, and a width parameter.
We also show how these parameters can be derived from a specific theory. This
self-similarity in the predictions can hopefully be used to search for
signatures of modified gravity on cluster scales in a model-independent way.
