You probably have exported your document as spreads instead of pages. Fortunately you can change that in CS6 (in CS5 you couldn´t)...
You can´t change that very easily in Acrobat, changing view settings won´t probably help because you already have 2 InDesign pages on every PDF page. Even if you choose to view single pages in Acrobat, you will see 2 pages at the time. I suggest that you re-export and change the setting marked with red triangle (on image below)
If you feel that re-exporting from InDesign is really hard, you can download handy little script for acrobat from:
If the first page in your document is a single page, check the comments. I have send a question about remaining the first page untouched, you can download a special version of that script which do that...you´ll find a link right after my comment...
I must warn you tough, script just duplicate pages and crop odd and even pages differently (you can also do that by yourself but it´s quite time consuming). Acrobat is notorius for its cropping method. Any page content will not be destroyed and most of the PDF processing systems (like flipping page converters) will ignore the cropping and show the whole spread anyway... so if it´s possible to re-export, I would do that.....
Thank you for your helpful and really detailed reply. As soon as I made the changes you recommended the document began to behave the way I wanted. As you pointed out I had selected to export as Spreads instead of Pages. The choices for view and layout I left as default and the exported documented came out just as I wanted it.
Thank you again for your help.