“Reduction” has a specific chemical meaning that is quite different from the common one. When iron rusts, the rust weighs more than the iron from which it came. That's because the rust contains all of the original iron plus the oxygen that came from the air. Iron ore is chemically identical to rust. When iron is extracted from its ore, the weight of the iron is less than that of the ore from which it came. That's because it has all of the original iron minus the oxygen that was present in the ore. Since the ore loses weight when turned into iron, chemists described the process as “reduction,” a word now used to mean the opposite of oxidation.
 
Another common term that has a specific meaning for chemists is “solubility.” In a solution, the dissolved material consists of individual molecules surrounded by water molecules. While a solution may be colored, it is always transparent. Examples of solutions include apple juice, wine, and tea. Translucent or opaque liquids like milk, orange juice, or hot chocolate are considered suspensions, not solutions. They consist of particles much larger than molecules, particles large enough to deflect light, and that is what makes them translucent or opaque.
 
What does all of this have to do with indigo? The indigo molecule actually has two forms. When the leaves of the indigo plant are fermented in water, atmospheric oxygen oxidizes the plant juices to form the deep blue, water-insoluble “indigotin,” C16H10N2O2 . The fact that it is insoluble in water is quite desirable for a clothing dye. After all, you don't want your new blue jeans to fade. But if indigotin is insoluble in water, how do you get it onto clothing in the first place? Historically, indigo dye was produced by soaking insoluble indigotin in fermenting urine. The bacteria in the urine consumed oxygen and converted insoluble blue indigotin to soluble, pale-green leucoindigotin, C16H10N2(OH)2. Cloth was placed into this brew, and when it was removed from the dye vat, atmospheric oxygen turned the dye back into insoluble blue indigotin. Figure 1 shows both forms of indigo. Note that where the reduced solution meets the air in the bottle, there is a layer of insoluble, oxidized indigo.
 
The only difference between these two forms is that oxygen is bonded to hydrogen in the soluble form, but not in the insoluble form. The two OH groups in leucoindigotin resemble water molecules, HOH, rendering it soluble in water. Indigotin is missing these two crucial hydrogen atoms, leaving it insoluble in water. Thus, indigotin makes a suspension in water, while leucoindigotin makes a true solution.
 
While the soluble, reduced form is needed for dyeing cloth, only the oxidized form is needed for coloring soap. After all, we want the color to stay in the soap, not leach into the soap dish. And when we wash our hands, we want the color to go down the drain, not bleed into our skin.
 
Indigo is sold in two commercial varieties. The traditional one is a dark blue powder that is insoluble in both oil and water. It can be mixed with a little oil to make a slurry and then added to your oil as with any other solid soap colorant. The newer variety has been pre-treated with lye. While it is sometimes described as “soluble” or “pre-reduced,” it is actually neither of those in the strict chemical sense. It forms an opaque suspension in water (not a transparent solution), and may be added to your lye portion. Either variety may be used in soap, but they should be used sparingly to avoid staining hands, sinks, and countertops. One gram of indigo per kilogram of oil produces a dark grayish-blue color in cold process soap.
 
Indigo is not the only vegetable colorant that can be used successfully in soap. Cocoa, coffee, and saffron all fare well. Other colorants change or even lose their color when exposed to the harsh alkalinity of raw soap. Each compound must be evaluated on a case-by-case basis, but if I had to come up with a rule of thumb, I would favor insoluble materials over soluble ones because the very features that make molecules soluble (like OH groups) also make them interact with acids and bases.
 
The most popular mineral soap colorants are oxides, micas, and ultramarines. The oxide pigments consist of metal oxides, whose colors depend on the identity of the metal, its oxidation state, and its state of hydration. Iron oxide, for example, comes in several varieties, from yellow, to red, to black. Micas are natural aluminosilicate minerals that consist of thin, interlocking sheets. They are naturally a pale, transparent yellow, but for soaps and cosmetics they are impregnated with dyes to give them vibrant colors. These dyes might react with raw soap, but being sandwiched between layers of mica, they are protected like pictures behind glass. Ultramarine is a complex mineral pigment produced from clay, sulfur, and soda ash.  A variety of colors is available, depending on its oxidation state.
 
The Indigo Swirls soap combines two blue colorants, both insoluble in water, and both used at the rate of 0.1% of the oil weight. The darker color is indigo, and the lighter one ultramarine blue. It's pretty Closer to Fine, if I say so myself.