f(g(x)) decreases while g(f(x)) increases
Python, 68 characters
f=lambda x:(1-x%2*2)*(2*x*x+(x&lt;0))
g=lambda x:(1-x%2*2)*(2*x*x+(x&gt;0))

f maps negative numbers to odd numbers and positive numbers to even numbers, and even numbers to positive numbers and odd numbers to negative numbers, with the output magnitude increasing strictly with the input magnitude.
g does the same, except it maps negative numbers to even numbers and positive numbers to odd numbers.
f ∘ g maps negative → even → positive and positive → odd → negative.
g ∘ f maps negative → odd → negative and positive → even → positive.
Therefore f and g have the desired properties.