I'd like to view the total number of upvotes on my questions, upvotes on my answers, downvotes on my questions, and downvotes on my answers. Is there any way I can do this without manually counting?
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I don't think there's anywhere in the main UI that shows that information. Your profile page shows the number of votes you've cast, but not the number you've received. I guess the idea here is that your reputation reflects that, albeit in a modified way.
You can get the raw numbers by doing a query on the Stack Exchange Data Explorer (linked as "data" in the footer of the SE sites). Note that it will be current as of the last monthly dump, not up-to-the-minute.
If you check the listing from http://stackoverflow.com/reputation you can see this.
The first column is the vote type, the second is the post id and the third is the reputation gained or lost.
So you can see that "1" is an acceptance - as it gives 15 reputation; "2" is an up-vote - you can see whether it's a question or answer from the amount of reputation you gain; "3" is a down-vote - in this case though you can't tell questions from answers.
There are other numbers which mean rep from edits to posts and tag-wikis, but I can't find that information right now.