  Generalized Quasitopological Gravities (GQTGs) are higher-order extensions of
Einstein gravity in $D$ dimensions satisfying a number of interesting
properties, such as possessing second-order linearized equations of motion on
top of maximally symmetric backgrounds, admitting non-hairy generalizations of
the Schwarzschild-Tangherlini black hole which are characterized by a single
metric function or forming a perturbative spanning set of the space of
effective theories of gravity. In this work, we classify all inequivalent GQTGs
at all curvature orders $n$ and spacetime dimension $D \geq 4$. This is
achieved after the explicit construction of a dictionary that allows the uplift
of expressions evaluated on a single-function static and spherically symmetric
ansatz into fully covariant ones. On the one hand, applying such prescription
for $D \geq 5$, we find the explicit covariant form of the unique inequivalent
Quasitopological Gravity that exists at each $n$ and, for the first time, the
covariant expressions of the $n-2$ inequivalent proper GQTGs existing at every
curvature order $n$. On the other hand, for $D=4$, we are able to provide the
first rigorous proof of the fact that there is one and only one (proper)
inequivalent GQTG at each curvature order $n$, deriving along the way a simple
expression for such four-dimensional representative at every order $n$.
