Effects of Inversion Centres
The coordinate effects of inversion centres are relatively easy to understand. As this fully inverts positions, an inversion centre placed at \((0, 0, 0)\) will move position \((x, y, z)\) to \((-x, -y, -z)\).
The position of the inversion centre works the same way as for mirror planes and rotation axes. The position of the inversion centre is doubled and added to the relevant coordinates. Consider the periodicity of the cell when doing this however.
An inversion centre at \((0, \frac{1}{4}, \frac{1}{2})\) thus moves position \((x, y, z)\) onto \((-x, -y + \frac{1}{2}, -z + 1)\). Of course, with periodic boundary conditions, this is identical to the position \((-x, -y + \frac{1}{2}, -z)\).