  In this paper we propose and investigate in full generality new notions of
(continuous, non-isometric) symmetry on hyperk\"ahler spaces. These can be
grouped into two categories, corresponding to the two basic types of continuous
hyperk\"ahler isometries which they deform: tri-Hamiltonian isometries, on one
hand, and rotational isometries, on the other. The first category of
deformations gives rise to Killing spinors and generate what are known as
hidden hyperk\"ahler symmetries. The second category gives rise to
hyperholomorphic line bundles over the hyperk\"ahler manifolds on which they
are defined and, by way of the Atiyah-Ward correspondence, to holomorphic line
bundles over their twistor spaces endowed with meromorphic connections,
generalizing similar structures found in the purely rotational case by Haydys
and Hitchin. Examples of hyperk\"ahler metrics with this type of symmetry
include the c-map metrics on cotangent bundles of affine special K\"ahler
manifolds with generic prepotential function, and the hyperk\"ahler
constructions on the total spaces of certain integrable systems proposed by
Gaiotto, Moore and Neitzke in connection with the wall-crossing formulas of
Kontsevich and Soibelman, to which our investigations add a new layer of
geometric understanding.
