Duotones are a graphic design mainstay with application to print media and textiles. In the case of mosaic images constructed with duotone tiles arranged in a grid, the "brightness" of each tile is tuned by adjusting the fractional areas within the tile occupied by a dark and a light color, respectively. Use of duotone tile types containing symmetries provides opportunities for algorithmic approaches to systematically representing pixel brightness, facilitating automation as an aid in constructing mosaic representations of images. A parametric representation using Bézier curves is described here for a type of Truchet-like square tiles each with a pair of opposite corners depicted in a contrasting color. A single parameter controls tile brightness by smoothly distorting curved boundaries between the color regions. Illustrations of the method include mosaics depicting simple gradients and more complex images. Additional considerations in some textile applications, including patchwork quilts and knitting, are also explored.