If there were compelling theoretical reasons, I might suppose that it existed. For example,

• if every particle had a charge that was an element of a particular group, which could be factored into the Cartesian product of four groups, one for each force, and
• a particle which has its charge being the identity element in any one of those groups doesn't feel that force, and
• this theory uses the group structure in some significant way, not just as a glorified table, and
• every element of the overall group has exactly one kind of particle with that exact combination of charges,
• except we couldn't tell whether there was a particle in the 'no interactions' slot because it didn't interact with anything...

I'd hazard that they exist, not that it would matter.