Some others have pointed out Sudarsky’s colors, but ignored that this is not the whole story:
The discrepancy [that neither Jupiter nor Saturn actually look like Sudarsky’s type I color] can be partially accounted for by taking into account non-equilibrium condensates such as tholin or phosphorus, which are responsible for the coloured clouds in the Jovian atmosphere, and are not modelled in the calculations.
So, a “non-equilibrium process” produces a strong color, and is confined to one hemisphere either because the colors vanish before it crosses the equator, and the process is associated with the location.
Consider that spots like the Great Red Spot can move around but can't cross tbe equator. Suppose that's true for air mass in general — the convection cells are separate for northern and southern. So, if something like a Big Red Spot was spewing tholins into the air, it would tint the entire hemisphere. But the mixing of air across the equator is much smaller in magnitude, and the color has a limited lifespan (either it disappears as molecules break down, or it returns to its proper deeper layer). So, it doesn’t color the whole planet.
Note that Saturn has an unusual feature at one pole. Something that exists at one pole only would be perfect for producing the color: either dreging it up from a deep layer, or producing molecules through some energetic process.
Another possibility for the latter would be the presence of a star bright in UV located in the north (or south) sky. This causes novel molecules to be produced where the star is visible. The slow production is balanced with the slow disintegration and slower cross-hemisphere mixing of the material.
A more exotic solution would be related to hyperfine splitting or shifting. Some chemical process happens slightly differently depending on whether it’s in a north or south magnetic field; this slight change gets magnified through shifting equilibrium etc.
Well, that seems difficult since the magnetic field alone won’t break the symmetry. But you don’t need anything, if the breaking is spontaneous. You have groups of interacting molecules that can settle into different possible equilibrium states. It happens to be in different states in north and south hemispheres, thanks to low mixing between them. It might be seasonal, with different temperature making the difference; so the summer and winter hemispheres will have characteristic colors.