I bought an inexpensive horizontal rod storage holder.  I didn't give it much thought but when I store my rods with the reel on, the rods have a slight bend in them.  Would this bend be enough to permanently damage my rods?

 

Notice how the rod on bottom has slight bend compared to rod (without reel) on top.

 

Not in the short term.  It's better to store them vertical for long term.

might help if you could store without the reel attached?  just a thought because the top one doesn't appear to have a reel nor a bend.

Graphite is much less apt to take a set especially if not subjected to high heat. Space the pieces to eliminate or minimize any sag and you should be fine. Vertical is not mandatory but is my preference for both care of the rods and utilization of space. 

Verticle with support is best, leaning against the garage wall/ corners of you're house, is a big no no

I lean my rods up against the wall in the corner of my garage and have never had any problems...

 

leaning against the garage wall/ corners of you're house, is a big no no

 

Not really. I have a few rods that don't fit in the rack, and I've leaned them up in the corner for years, with no ill effects.

I had two all star 7' popping rods that took a set in the mid 90's , I made that post based on my experience and no other reason.

As long as they aren't bent, and exposed to heat, they're fine.

I use those also, have it set up in my garage. Mine holds six rods. The rods do bend a little with the reels on. I don't worry about it to much because I use the rods all the time, and have not seen them set. In The winter the rods come down, the reels come off, and the rods go in a PVC tube that stands vertically in my basement. I have a question: My garage is separate from the house, and has no heat or ac, if I store my rods in there for the winter the same way I store them in the basement would the cold affect the rods? I live in New Jersey.

Not really. I have a few rods that don't fit in the rack, and I've leaned them up in the corner for years, with no ill effects.

 X3

Even if they are exposed to heat, they'll be fine. Unless you're buying from some knock-off Chinese company, then the prepreg has already been cured. You can't cure resin twice.