I bet that the answer to my question is simply yes, but I'm unsure. Consider Example 20.3 in Maschler, Solan & Zamir (pages 803 & 804). In that example, four vectors are presented: namely, \begin{align} \theta(\mathbf{x})&=(1/3,1/3,0,0,-1/3,-1/3,-1/3,-2/3)\\ \theta(\mathbf{y})&=(0,0,0,0,0,0,0,-1)\\ \theta(\mathbf{z})&=(1/3,1/6,0,0,-1/6,-1/3,-1/2,-1/2)\\ \theta(\mathbf{w})&=(1/3,1/3,0,0,0,-1/3,-1/3,-1)\\ \end{align}
Clearly, the elements of the vectors above are written in decreasing order. When comparing these four vectors lexicographically in this particular order, we conclude that $\theta(\mathbf{y})<_L\theta(\mathbf{z})<_L\theta(\mathbf{x})<_L\theta(\mathbf{w})$.
My question is simple: is it possible to order the elements within each vector in a different manner and still compare them lexicographically? Or is this decreasing ordering a necessary requisite to compare them lexicographically?
I think the answer to your question is "no".
You can compare lists of numbers like these lexicographically whether or not they are decreasing. The meaning of "lexicographic" is that if the coordinates match in the first place you compare the second place elements, and so on. If you reorder the coordinates in the items that will change the order in which they appear lexicographically.