Using Brian Sanderson's Pattern Recognition Algorithm (link) we readily see that the pattern below corresponds to the Wallpaper Group labelled p4m. This is just an infinite checkerboard of alternating black and white squares. Notice with this pattern that if we rotate the whole checkerboard by 90 degrees around the center of a square the plane is unchanged. Black squares are mapped to black squares and white squares are mapped to white squares. However, if we rotate the whole checkerboard by 90 degrees around the vertex of a square, black squares are mapped to white squares (and vice-versa). Is this a 'colour reversing symmetry'? If so, does it belong to the symmetries of p4m? Or does it belong to some larger group that includes both colour preserving and colour reversing symmetries? (As Frank Farris seems to be suggesting on page 123 of his book 'Creating Symmetry'). Notice however, that black and white squares are also reversed if we shift the whole checkerboard by one square (up/down or left/right), which suggests that the colour reversing symmetry is also a symmetry (because the plane is infinite and so shifts yield the same pattern). Now I'm confused. Can someone help me out? (By the way, I do understand that when defining the symmetry of a coloured wallpaper pattern in addition to having an isometry we also have a function that maps points to colours).
I agree that the term "wallpaper pattern" is usually tossed around without giving a rigorous definition, and this can lead to confusion. One workable definition is that it is a periodic function defined on the plane. The function may be real-valued if you wish to create gray-scale images or complex-valued if we wish to include color as Frank Faris does).
The color wheel can be modeled using the complex plane. Each point on the unit circle has a color or complex phase value associated to it, and radial distance inn the plane describes saturation. A colored wallpaper pattern is then a periodic function $f: {\mathbb R}^2\to {\mathbb C}$.
Within a given fundamental period parallelogram, a colored pattern may have additional color-reversing symmetries (quasi-symmetries). Such quasi-symmetries are Euclidean motions $g$ for which $ f(g(x)) = m(g) f(x)$ where the complex multiplier $m(g)$ represents a phase-shift that alters the color. Since iterates of $g$ correspond to repeated alterations of phase, the phase changes must be cyclic. Thus it turns out the multipliers take values on the unit circle $U$ in the complex plane. The group of quasi-symmetries is richer than the collection of true symmetries. (The true symmetry group is a subgroups of the group of quasi-symmetries.)
P.S. Since $f$ is periodic, it has two linearly independent fundamental period vectors. Taking $x,y$ to be coefficients of these basis vectors, we can WLOG assume that the fundamental period parallelogram is the unit square. Then $f$ can be expressed as a complex Fourier series in two variables: $f(x,y) = \sum_{j,k} c_{j,k} e^{ 2\pi i (k x+jy)}$. The frequency lattice $(j,k)$ carries a collection of complex-valued coefficients $c_{j,k}$ that uniquely determine $f$. Thus we can also regard the wallpaper pattern as an assignment of complex numbers on the frequency lattice. Frank Farris discusses in detail how affine motions acting in $(x,y)$ space such as $(x,y)\to (x',y')=(ax+by +t, cx+dy + s)$ manifest themselves as phase-change factors on the frequency lattice. The frequency-plane description sheds extra light on what possible discrete subgroups are possible. In particular one sees easily that e.g translation through a fraction of a full period must correspond to a phase shift through a fraction of the unit circle.