Place the red cabbage and beets in separate small pots and cover with water. Bring to a boil, reduce heat to low and simmer for 20 minutes. Strain the liquid and cool.
Add 1 tablespoon of lemon juice to the cabbage water to get violet eggs or 1 tablespoon of baking soda to get bright blueish green eggs.
Allow the beets to soak in their cooking liquid (off the heat) for 1 hour to get red, or use immediately for a rust-ish purple color.
Place the egg whites in the color mixtures to dye. The egg whites will begin to absorb the color immediately and will continue to darken and become more vibrant over the next 20 minutes.
To dye whole, unpeeled eggs, add 1 tablespoon of vinegar to each color. The eggs will absorb the color after about 10 minutes and will continue to darken and become more vibrant up to 24 hours in the liquid. If you will be soaking the eggs overnight or several hours, cover and refrigerate them.
Place the eggs in a pot and cover with cold water. Bring the water and eggs to a boil over high heat. When the water comes to a boil, turn off the heat and let sit for 12 minutes.
Drain off the hot water and cover with cold water to cool the eggs.
Peel the eggs and cut in half for deviled eggs. Leave the eggs whole and unpeeled to dye the egg shells.
In a bowl, mash together the egg yolks, mustard, mayonnaise, and salt until smooth. Fill the halved egg whites with the mixture and serve.
