[CentOS] Email access via Android device

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?

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