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Let L = (L-1 , L-2) be a list consisting of a sublist L(1 )of powers of irreducible (monic) scalar polynomials over an algebraically closed field F, and a sublist L-2 of nonnegative integers. For an arbitrary such list L, we give easily verifiable necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some T-palindromic quadratic matrix polynomial with entries in the field F. For L satisfying these conditions, we show how to explicitly construct a T-palindromic quadratic matrix polynomial having L as its structural data; that is, we provide a T-palindromic quadratic realization of L. Our construction of T-palindromic realizations is accomplished by taking a direct sum of low bandwidth T-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of T-palindromic quadratic polynomials is that all even grade T-palindromic matrix polynomials have a T-palindromic strong quadratification. Finally, using a particular Mobius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with T-even structure.