  We introduce the notion of tame $\rho$-quaternionic manifold that permits the
construction of a finite family of $\rho$-connections, significant for the
geometry involved. This provides, for example, the following: (1) a new simple
global characterisation of flat (complex-)quaternionic manifolds, and (2) a new
simple construction of the metric and the corresponding Levi-Civita connection
of a quaternionic-K\"ahler manifold by starting from its twistor space;
moreover, our method provides a natural generalization of this correspondence.
Also, a new construction of quaternionic manifolds is obtained, and the
properties of twistorial harmonic morphisms with one-dimensional fibres from
quaternionic-K\"ahler manifolds are studied.
