Let T_θ and T_φ{symbol} be C₁₀ contractions with characteristic functions θ∈H^∞ (ℂⁿ→ℂⁿ⁺¹), φ{symbol}∈H^∞ (ℂ^m→ℂ^m+1). The fundamental result is: T_θ and T_φ{symbol} are quasisimilar if and only if[Figure not available: see fulltext.] The paper contains an analysis of this condition; examples are given.