  We first construct an action of the extended double affine braid group
$\mathcal{\ddot{B}}$ on the quantum toroidal algebra
$U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in untwisted and twisted types. As a
crucial step in the proof, we obtain a finite Drinfeld new style presentation
for a broad class of quantum affinizations. In the simply laced cases, using
our action and certain involutions of $\mathcal{\ddot{B}}$ we produce
automorphisms and anti-involutions of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$
which exchange the horizontal and vertical subalgebras. Moreover, they switch
the central elements $C$ and $k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$ up to inverse.
This can be viewed as the analogue, for these quantum toroidal algebras, of the
duality for double affine braid groups used by Cherednik to realise the
difference Fourier transform in his celebrated proof of the Macdonald
evaluation conjectures. Our work generalises existing results in type $A$ due
to Miki which have been instrumental in the study of the structure and
representation theory of $U_{q}(\mathfrak{sl}_{n+1,\mathrm{tor}})$.
