  Noether's theorems are widely praised as some of the most beautiful and
useful results in physics. However, if one reads the majority of standard texts
and literature on the application of Noether's first theorem to field theory,
one immediately finds that the ``canonical Noether energy-momentum tensor"
derived from the 4-parameter translation of the Poincar\'e group does not
correspond to what's widely accepted as the ``physical'' energy-momentum tensor
for central theories such as electrodynamics. This gives the impression that
Noether's first theorem is in some sense not working. In recognition of this
issue, common practice is to ``improve" the canonical Noether energy-momentum
tensor by adding suitable ad-hoc ``improvement" terms that will convert the
canonical expression into the desired result. On the other hand, a less common
but distinct method developed by Bessel-Hagen considers gauge symmetries as
well as coordinate symmetries when applying Noether's first theorem; this
allows one to uniquely derive the accepted physical energy-momentum tensor
without the need for any ad-hoc improvement terms in theories with exactly
gauge invariant actions. $\dots$ Using the converse of Noether's first theorem,
we show that the Bessel-Hagen type transformations are uniquely selected in the
case of electrodynamics, which powerfully dissolves the methodological
ambiguity at hand. We then go on to consider how this line of argument applies
to a variety of other cases, including in particular the challenge of defining
an energy-momentum tensor for the gravitational field in linearized gravity.
Finally, we put the search for proper Noether energy-momentum tensors into
context with recent claims that Noether's theorem and its converse make
statements on equivalence classes of symmetries and conservation laws$\dots$
