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.

Real mirror:

.mirror{
    display: inline-block; 
    font-size: 30px;

    -webkit-transform: matrix(-1, 0, 0, 1, 0, 0);
    -moz-transform: matrix(-1, 0, 0, 1, 0, 0);
    -o-transform: matrix(-1, 0, 0, 1, 0, 0);
    transform: matrix(-1, 0, 0, 1, 0, 0);
}

<span class='mirror'>Mirror Text<span>

You can user either

.your-class{ 
      position:absolute; 
      -moz-transform: scaleX(-1); 
      -o-transform: scaleX(-1); 
      -webkit-transform: scaleX(-1); 
      transform: scaleX(-1); 
      filter: FlipH;  
}

or

 .your-class{ 
  position:absolute;
  transform: rotate(360deg) scaleX(-1);
}

Notice that setting position to absolute is very important! If you won't set it, you will need to set display: inline-block;