The half-linear difference equations with the deviating argument Δ ( a n | Δ x n | α sgn   Δ x n ) + b n | x n + q | α sgn   x n + q = 0 , q   ∈   ℤ are considered. We study the role of the deviating argument q , especially as regards the growth of the nonoscillatory solutions and the oscillation. Moreover, the problem of the existence of the intermediate solutions is completely resolved for the classical half-linear equation ( q = 1). Some analogies or discrepancies on the growth of the nonoscillatory solutions for the delayed and advanced equations are presented; and the coexistence with different types of nonoscillatory solutions is studied.