Yum says a package is installed, and not installed, and it's blocking other packages from being installed

The RPM DB is confused. My recommendation, having encountered this before, it to rpm -e --justdb the newer "duplicate" package installed, then yum upgrade the package, then rpm --rebuilddb just because I'm a bit of a paranoiac. Once you've gone through the first two steps, you should be able to install the newer packages, assuming your repo is properly in sync. You may want to yum clean all right after the rpm -e --justdb to be sure of that - and make darn sure you specify --justdb or you could do all sorts of bad things to your system.

Messing with the RPM database didn't yield any particularly good results.

I ended up noticing that some of the glibc packages were i686 and others were x86_64. For instance:

Package: glibc-2.12-1.149.el6.i686 (CentOS-OS)
    Requires: glibc-common = 2.12-1.149.el6
    Installed: glibc-common-2.12-1.149.el6_6.5.x86_64

I didn't like that, and I expect that Yum didn't like that either.

Running

yum downgrade glibc glibc-headers glibc-common glibc-devel

brought all of the packages to the same architecture (x86_64). Then,

yum install glibc-static

worked like a charm.