I found the answer, written by Claudette M., in another post. Here is what
she said to do, and it works!
The only way I found to solve the problem of text not rotating along with the
autoshape is to fool Word into believing that the shape is now a picture; you
can rotate pictures. First, on a blank page create your autoshape, insert
your text and do all the formating you need. Then copy the autoshape.
Return to the document where you want to put the object, and click Edit,
Paste Special, Picture and you've now inserted a "picture" on the page. Grab
the corner and stick it wherever you need it and you can format size, etc.
(as a whole - not individual components of the autoshape) through the format
picture menu.