It works one way, but not the other. You can use the keys gained in Steam (Right click on game in left-hand menu, View Game CD Key) and activate them for your EA Games account by registering them at https://activate.ea.com, which makes them then appear in your Origin games list for installation. (This applies to most - but not all - EA Games that are available on Origin)
You cannot use Origin codes to apply games to Steam, and Steam won't let you buy DLC for games you don't own though Steam.
[2015 update: You now activate them on Origin though the client. Origin menu, Redeem Product Code.]
I know of only one way that this can happen in windows, and has nothing to do with Steam other than their not-so-great decision to install all content into Program Files.
If you are on Windows 7 or Windows Vista, and you have UAC set to ON, then change it later to OFF, this can happen.
UAC does thing evil thing where if a program tries to write to a "protected" area like Program Files, EVEN IF YOU ARE ADMINISTRATOR, when you get a UAC prompt and give permission, it actually writes any files that would normally go into that directory into a "virtualized" directory somewhere else.
If you turn off UAC, that will no longer happen, and as an awesome bonus, it will no longer know about the files that it virtualized.
So in your case, if you had UAC on, installed a ton of games, then later got sick of UAC and turned it off, this would happen as you described.
See the second paragraph in "Features" in the article on User Account Control.
If you have done this and turn on UAC again, your content will come back, but be a TOTAL mess because if you've downloaded more stuff with UAC off, then that will be invisible when you turn UAC on, and vice versa.
this drove me a little crazy once before I realized what was happening.
i'm curious to know if this is your issue.
Best Answer
You can try redeeming your keys (official instructions here) first. If you can redeem them there, you will not need to repurchase everything on Steam. However, if you cannot redeem the keys on Steam, you almost certainly would need to repurchase all of the DLC on steam as well.