There's an interesting combination of dyes being sold under the name AlterEgo. A dye that colors only silk is combined in a single dyebath with a dye that colors both silk and rayon. The result is a contrast in color between the silk backing and the rayon plush, in silk/rayon velvet scarves, especially visible if areas of the rayon have been eaten away by devoré.
AlterEgo dyes are very expensive, compared to other dyes. Each small bottle of color costs $14, and you have to buy two colors at once, one for the silk and one for the rayon, plus a special dye fixative, whose use is not optional, for $6.50. The direct dye is only poorly washfast, so the proprietary fixative has to be used so that it doesn't wash out.
There are other ways to get similar effects. The easiest and least expensive way is to use a pre-mixed Procion MX dye color, with soda ash, in a single step. Since pre-mixed dye colors tend to produce different hues on protein fibers than on cellulose, you can get a good contrast this way. We need to study which dyes come out stronger on silk, and which weaker, to be able to predict our results....
Another way is to choose your own dyes to mix like the AlterEgo dyes. You can buy acid dyes and direct dyes separately from many sources. Acid dyes work only on silk, but some will also stain rayon. All of the direct dyes I know about will color both rayon and silk, but they do not color the silk quite as dark. Direct dyes have the advantage of working in the same bath as acid dyes, but they are not very good dyes, in that they tend to wash out. You can use a dye fixative to glue that dye into the fabric, or you can use a single-color Procion MX dye instead, in a separate step from the acid dyeing.
If you are double-dyeing with acid dyes and direct dyes, but not at the same time, which dye should you aply first? Always apply the acid dye first, if you want to dye the two fibers different colors. Some acid dyes are also in the category of dyes known as basic dyes, and will stick to the direct dye, even if they cannot stick to the cellulose fiber.