  For a graph class ${\cal H}$, the graph parameters elimination distance to
${\cal H}$ (denoted by ${\bf ed}_{\cal H}$) [Bulian and Dawar, Algorithmica,
2016], and ${\cal H}$-treewidth (denoted by ${\bf tw}_{\cal H}$) [Eiben et al.
JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the
"torso" of the graph induced on a modulator to the graph class ${\cal H}$.
Here, the torso of a vertex set $S$ in a graph $G$ is the graph with vertex set
$S$ and an edge between two vertices $u, v \in S$ if there is a path between
$u$ and $v$ in $G$ whose internal vertices all lie outside $S$.
  In this paper, we show that from the perspective of (non-uniform)
fixed-parameter tractability (FPT), the three parameters described above give
equally powerful parameterizations for every hereditary graph class ${\cal H}$
that satisfies mild additional conditions. In fact, we show that for every
hereditary graph class ${\cal H}$ satisfying mild additional conditions, with
the exception of ${\bf tw}_{\cal H}$ parameterized by ${\bf ed}_{\cal H}$, for
every pair of these parameters, computing one parameterized by itself or any of
the others is FPT-equivalent to the standard vertex-deletion (to ${\cal H}$)
problem. As an example, we prove that an FPT algorithm for the vertex-deletion
problem implies a non-uniform FPT algorithm for computing ${\bf ed}_{\cal H}$
and ${\bf tw}_{\cal H}$.
  The conclusions of non-uniform FPT algorithms being somewhat unsatisfactory,
we essentially prove that if ${\cal H}$ is hereditary, union-closed,
CMSO-definable, and (a) the canonical equivalence relation (or any refinement
thereof) for membership in the class can be efficiently computed, or (b) the
class admits a "strong irrelevant vertex rule", then there exists a uniform FPT
algorithm for ${\bf ed}_{\cal H}$.
