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Let H be a complex, separable Hilbert space and let B(H) denote the algebra of all bounded linear operators acting on H. Given a unitarily-invariant norm |⋅|u on B(H) and two linear operators A and B in B(H), we shall say that A and B are polynomially isometric relative to |⋅|u if |p(A)|u = |p(B)|u for all polynomials p. In this paper, we examine to what extent an operator A being polynomially isometric to a normal operator N implies that A is itself normal. More explicitly, we first show that if |⋅|u is any unitarily-invariant norm on Mn(C), if A, N ∈ Mn(C) are polynomially isometric and N is normal, then A is normal. We then extend this result to the infinite-dimensional setting by showing that if A, N ∈ B(H) are polynomially isometric relative to the operator norm and N is a normal operator whose spectrum neither disconnects the plane nor has interior, then A is normal, while if the spectrum of N is not of this form, then there always exists a nonnormal operator B such that B and N are polynomially isometric. Finally, we show that if A and N are compact operators with N normal, and if A and N are polynomially isometric with respect to the (c,p)-norm studied by Chan, Li, and Tu, then A is again normal. [ABSTRACT FROM AUTHOR]