I'm pretty sure that this is possible, but I don't currently know enough about email to know where to start.
My main desktop computer runs Centos 6 and my preferred email client is Sylpheed, which supports both POP and IMAP email, and my "internal network" has a static IP address, so getting access to the computer from the big scary world via my phone isn't a problem.
I have six different email accounts that live on various servers. I have Sylpheed set up to poll each account once every ten minutes using POP3 and download all incoming mail. Upon receipt, I have a whole lot of filters in Sylpheed to sort the mail into various subdirectories of my Mail directory. For example, one of my directory structures under Mail is inbox/1/2, and email may be sorted in Mail/inbox or Mail/inbox/1 or Mail/inbox/1/2, or other subdirectories outside of inbox like, for example, Mail/3. In short, I have a dozen or so subdirectories that Sylpheed sorts mail into based on various criteria like From or Subject.
What I would like to do is somehow make this whole thing available on my Android phone (Samsung Galaxy 3). As it sits right now, if I am not sitting in front of my computer the only way I can check my email is to ssh into it and look at the files in each of those subdirectories using a text utility like less; this is really inconvenient and I can't reply to emails that way until I get back in front of my computer.
I'm thinking that I need to run some kind of a mailserver on my computer that can be accessed by both Sylpheed locally and by ??? on my Android device. I get the impression that if I wanted to hand all of my email over to gmail I could then have something like this working. I don't particularly want to do that; I would rather have something running on my own machine to do it.
I could use something else to poll for my email (fetchmail?). Sylpheed does have a configuration setting that I'm not currently using that says "Enable strict checking of summary caches -- enable this if the contents of the folders may be changed by other applications" so based on that I think it can handle having messages magically changed by a program on my Android device.
What is the best way to approach this? My best idea so far is to set up fetchmail (or something) to do the pop downloads of incoming mail, and have some kind of a local imap server running though which I access the actual mail via any email client that can work with an imap account. Then I can set up Sylpheed to access only one account, that being the one on my local computer, and run some imap-using mail client on the Android device for remote access.
Having never actually used an imap email account in this way, that leads to a couple of other questions. (The only thing I've ever done with imap is set up squirrelmail.) What about filtering the email into those directories? Procmail? And what happens to sent email -- I assume that the SMTP part of the mail client wouldn't actually change -- I would still send outbound mail directly through the mailserver where my email account exists, right? But does it (can it) also send a copy to the imap mail store so I can send email from Android and later on review what I sent with Sylpheed?