Symbolism, Imagery, Allegory
Mirror, Mirror On The Wall
Here's the deal: Eliza has never looked at herself in a mirror, and she doesn't want to start making a habit of it. This following exchange is super-odd for us citizens of the 21st Century:
HIGGINS. What was wrong, Mrs. Pearce?
MRS. PEARCE [blandly] Oh, nothing, sir. It doesn't matter.
LIZA. I had a good mind to break it. I didn't know which way to look. But I hung a towel over it, I did.
HIGGINS. Over what?
MRS. PEARCE. Over the looking-glass, sir.
HIGGINS. Doolittle: you have brought your daughter up too strictly.
DOOLITTLE. Me! I never brought her up at all, except to give her a lick of a strap now and again. Don't put it on me, Governor. She ain't accustomed to it, you see: that's all. But she'll soon pick up your free-and-easy ways.
LIZA. I'm a good girl, I am; and I won't pick up no free and easy ways. (2.303-313)
Yes. Liza, ever the "good girl," won't look at herself in the mirror.
She's accustomed to living without mirrors and—conveniently enough—is accustomed to thinking that looking in mirrors is both an act of extreme vanity and probably a little lewd. So when she's confronted with her first mirror, she drapes a cloth over it.
This is one of the first moments we see Liza's strength of character come out. She refuses to indulge in the habits of the rich that she thinks are unbecoming or unladylike. Her constant refrain of "I'm a good girl, I am" is more important to her than succumbing to pleasure or fashion.
We see this play out in a number of ways as Pygmalion continues, most notably in the fact that she refuses to marry for money. Marrying for money was commonplace in the posh London circles of the early 20th Century, but to Liza it smacks of prostitution. She wouldn't have dreamed of marrying for money when she was a flower-seller, and she's not about to now that she has a new accent and some fancy new clothes.
Her education and elocution lessons may have given her some freedom, but she's not free-and-easy.