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The half-linear difference equations with the deviating argument Δ(an|Δxn|αsgn Δxn)+bn|xn+q|αsgn xn+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.