You got your single and double quotes backwards.
Looks like you need to say:

$aword          = 'foo';     # use single quotes instead of double 
$containmentvar = "$aword";  # use double quotes instead of single
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Variables between double quotes will be interpolated to their values, but between single quotes they will be taken literally.

Hope this helps,
Jeff

UPDATE:
Hmm, when I first read your post, I thought you were wanting symbolic references - I have to agree with tilly on this one, and do read ar0n's link that he provided, but here you go anyways:

$aword           = 'foo';
$containmentvar  = 'aword';  # no dollar sign here
$$containmentvar = 'bar';    # foo or bar?
print "$aword\n";
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Remember, you can't use strict with this code, that should be reason enough to find another solution, but if you aren't coding heart monitoring devices you should be ok.

Now that I have handed a loaded gun to somebody, I am going to find a lawyer.