I think modern labour might be too diverse to cover properly with only a few symbols. We're no longer uniting two labour forces, material conditions have changed. I think that instead of focusing on types of labour, we might instead focus on what the movement can bring.

We could put basic human rights on a theoretical new flag. A drop represents access to water, bread represents access to food, and a roof over both.

If you want a new flag, picking something that unites is all (our needs) might be a good way to go.