May be due to the symmetry of the matrix M the order of operation makes no difference. I am not sure if this identity holds if M is not symmetric.

A symmetric matrix is a square matrix that satisfies AT = A, where AT denotes the transpose. Next, in general it holds (AT)T = A. Finally, for the product of two matrices A and B it holds:

(AB)T = BTAT (see: http://mathworld.wolfram.com/Transpose.html),

therefore we have:

(X0T⋅M⋅X)T = (X0T⋅(M⋅X))T = (M⋅X)T⋅(X0T)T = XT⋅MT⋅X0 = XT⋅M⋅X0

Anton Vrdoljak has answered the question. Just to add a minor comment: X0T⋅M⋅X is a scalar (if X and X0 are column vectors), therefore, X0T⋅M⋅X = (X0T⋅M⋅X)T