We consider possible conformal field theory (CFT) descriptions of the
various inertial ranges that exist in 2d duality-invariant magnetohydr
odynamics. Such models arise as effective theories of dyonic plasmas i
n 3 dimensions in which all fields are independent of the third coordi
nate. We find new constraints on the allowed CFTs compared to those th
at may describe turbulence in 2d plasmas of electric charges only. The
predictions from CFT concerning equipartition of energy amongst the e
lectric and magnetic fields are discussed, and quantities exhibiting u
niversal scaling are derived.