What about this:
var result = random.Next(0, 2) * 2 - 1;
To explain this, random.Next(0, 2)
can only produce a value of 0 or 1. When you multiply that by 2, it gives either 0 or 2. And simply subtract 1 to get either -1 or 1.
In general, to randomly produce one of n integers, evenly distributed between a minimum value of x and a maximum value of y, inclusive, you can use this formula:
var result = random.Next(0, n) * (y - x) / (n - 1) + x;
For example substituting n=4, x=10, and y=1 will generate randum numbers in the set { 1, 4, 7, 10 }.