Actually there are many ways how to cycle a palette on NES.
You can cycle colors 1,2,3 in every palette (like, 2 goes to 1, 3 to 2, 1 to 3) - this keeps the background color the same and uses the same predefined set of colors. You can cycle all four colors (1 to 0, 2 to 1, 3 to 2, 0 to 3), this will change the background color too, but don't forget that it is shared between all palettes, so only last write to any color 0 will take effects - thus the background will always have one of colors of one of palettes.
You can also cycle colors between palettes, cycle the 4-color sub palettes itself (all colors of palette 1 to palette 0 etc).
You can even cycle color values. Notice that low nibble of a hex code for a color is hue (0-d), and high nibble is brightness (0-3). So you can keep the top nibble while incrementing or decrementing low nibble, and have effect similar to the one that you can see while moving the Hue slider in a graphics editor.