What's wrong with the MHL connector (for HDMI output) on the Samsung Galaxy S3

MHL is an emerging standard for connecting mobile devices to HD displays. A lot of new smartphones support MHL via a dual-purpose connector that supports both micro-USB cables and MHL-HDMI adapters. Samsung advertises the Galaxy S3 as having one of these micro-USB / MHL connectors. The micro-USB works great, but my MHL-HDMI adapter doesn't work with it.