Can you use CSS to mirror/flip text?

You can use CSS transformations to achieve this. A horizontal flip would involve scaling the div like this:

-moz-transform: scale(-1, 1);
-webkit-transform: scale(-1, 1);
-o-transform: scale(-1, 1);
-ms-transform: scale(-1, 1);
transform: scale(-1, 1);

And a vertical flip would involve scaling the div like this:

-moz-transform: scale(1, -1);
-webkit-transform: scale(1, -1);
-o-transform: scale(1, -1);
-ms-transform: scale(1, -1);
transform: scale(1, -1);

DEMO:

span{ display: inline-block; margin:1em; } 
.flip_H{ transform: scale(-1, 1); color:red; }
.flip_V{ transform: scale(1, -1); color:green; }

<span class='flip_H'>Demo text &#9986;</span>
<span class='flip_V'>Demo text &#9986;</span>

-moz-transform: scale(-1, 1);
-webkit-transform: scale(-1, 1);
-o-transform: scale(-1, 1);
-ms-transform: scale(-1, 1);
transform: scale(-1, 1);

The two parameters are X axis, and Y axis, -1 will be a mirror, but you can scale to any size you like to suit your needs. Upside down and backwards would be (-1, -1).

If you're interested in the best option available for cross browser support back in 2011, see my older answer.