I propose a new blend mode called Rotate Hue (or Hue Shift.) It's purpose is to use the current layer to add or subtract the hue value of the "bottom" layers.

Calculation

A = Pixel of a layer with Rotate Hue blend mode
I = Input pixels.
[.h, .s, .v] = HSV values of a pixel with ranges [0deg - 360deg, 0% - 100%, 0% - 100%], all normalized to 0.0 - 1.0 for easier calculation. The hue (.h) component wraps around at 0.0 and 1.0.

O = Output pixel

O.h = A.v - 0.5 + I.h
O.s = I.s
O.v = I.v

In plain English terms:

[*] Pure 50% gray has no effect.
[*] 75% gray (light gray) shifts the hue of the bottom layers 90 degrees. Red #F00 is transformed to a bright yellowish green #80FF00
[*] 25% gray (light gray) shifts the hue of the bottom layers -90 degrees. Red #F00 is transformed to a vivid purple color #8000FF
[*] Pure white shifts the hue of the bottom layers 180 degrees. Red #F00 is transformed to Cyan/Aqua #00FFFF
[*] Pure black shifts the hue of the bottom layers -180 degrees. Red #F00 is transformed to Cyan/Aqua #00FFFF
[*] The hue and saturation of the blending layer is ignored.

Purpose

It can be used to make effects such as red-shifting and blue-shifting in the same layer much easier.

Difference from Hue and Overlay blend modes

The Hue blend mode sets the hue value of the bottom layers. Rotate Hue changes the hue value by a specified amount.

Overlay can do similar effects, though Overlay is less predictable.