If I subscribe to digital cable, can my analog tuners still pick up analog channels?

Hello, this all started because I'm shopping for internet, and Time Warner is offering me a deal where I can bundle internet and digital cable TV together at a price equal to what I currently pay for analog cable (about $61 which includes taxes) + about $26 for roadrunner (cable based internet, substantially faster than the DSL I've been using).

My wife is actually totally happy with the analog cable we currently get from Time Warner, and to get the same price as analog, I'd have to not get a converter box, as that would add another $8 per month. The sales lady tells me I can plug in the coax to an analog tuner, and still get all the same analog cable channels that we're currently getting with our analog subscription. This sounds kind of fishy to me, so I was wondering if you guys here could confirm if what she's telling me makes sense.

By the way, even if I were to fork out the extra $8 / month for a converter box, my wife doesn't watch live TV anyhow; she watches TiVo, and it's one of the older S2 models of TiVo, meaning it has a built in analog tuner - no digital tuner. The only way a converter box would be of use to her is if there is someway to get the TiVo to control it (which there might be a way, I don't know.)

But she really doesn't care about the extra digital channels anyhow, so I'd be interested to know if the digital signal coming in would really have the analog signal buried within it, so that her analog TiVo could still tune into it.

Also, by the way, my TiVo I use is an HD TiVo, which will tune into digital cable if I were to install a cable card, which Time Warner will rent to me for $2.99 / month. And since I have an HD TV to watch it on, I might be interested to rent the card, so this digital-for-the-same-price as analog kind of appeals to me.

Thanks for all these informative answers. It does seem like this this is the best forum I know to pose my audio/video questions and get such quick responses.

It also occurs to me, after speaking to my brother-in-law who also subscribes to a bundle with Time Warner (his bundle is high speed internet + digital TV + phone, but my bundle wouldn't include the phone) that his price is way higher than mine (even taking his phone into account), so I wonder if this offer I got is a promotional deal, and maybe 12 months from now, they're going hit me with a significant price increase. I suppose even if that's true, it's worth doing this for now, and 12 months from now, if I want to shop around again, I can.