  We construct conformally scalar NUTty dyon black holes in non-linear
electrodynamics (NLE) which possess conformal and duality-rotation symmetries
and are characterized by a free dimensionless parameter. The thermodynamic
first law of the black hole is formulated. Then we explore the strong
gravitational effects of the black hole, mainly focusing on the innermost
stable circular orbits (ISCOs) of massive particles and shadows formed by the
photons. One of our interesting results is that if the black hole is endowed
with a positive real conformal scalar charge rendering it
Reissner-Nordstr\"om-like, the radii of the ISCOs and the shadow both increase
with the increasing NLE parameter signifying the increasing nonlinearity of the
electromagnetic field.
