1.622 is the other side of the Java 7 switch.
As a result there are classes in core that have been compiled with Java 7 bytecode and references to methods / classes that are only available in Java 7.
While you can get the remoting connection up - as that does not rely on classes from Jenkins itself - running a build will be a different problem entirely.
You probably need to switch back to 1.609 LTS as that is the last LTS that is compiled against Java 1.6.
You will also need to be careful with the plugins that you use. Plugins can send code to the remote slaves. If they try to send some Java 1.7 code to your slaves running 1.6... boom!