Consider:

Grid[
 Table[Row[{i, j}], {i, 10}, {j, 10}],
 ItemSize -> {3, 2},
 Dividers -> All,
 Alignment -> {
     {Left, {Left, Right, Center}},
     {Top, Top, {Top, Center, Bottom}}}
 ]

The horizontal alignments go:

Left | Left Right Center | Left Right Center | Left Right Center 

The vertical bars signal where the cyclic alignments (re)start. The first Left is because of the explicit alignment specified before the cycle indicated by {Left, Right, Center}. The vertical alignments are parsed similarly. The main difference is that the alignments of the first two rows are specified before the cycle. The vertical alignments go:

Top Top | Top Center Bottom | Top Center Bottom | Top Center

They end mid cycle at the 10th row.

I believe the alignment setting you want is

Alignment -> {{{Left, Center, Right}}, {{Top, Center, Bottom}}}

However, given all of the other options you've provide, it will be difficult to actually see the alignment effect. The Spacings option doesn't change the size of the content area of the cells, just the spacing between the content areas. Replace the Spacings option with

ItemSize -> {10, 5}

and you'll be able to see the alignments.

You are right, Hans Milton. I had not seen that tech note. I didn't realize until I read your post, that it is one of the documents with a link in the Alignment documentation.

Well done, Michael Rogers. Your explanation with its well-chosen example is especially helpful. In my opinion, your example should be added to the Alignment documentation. Contrasting the cyclic arguments with the non-cyclic ones is an effective way to illustrate how to use cycling, and your explanation eliminated any ambiguity remaining in the example.