Does polarity matter for a DC rechargable battery?
Absolutely YES ...the polarity does matter.

Even though your cable ma not indicate any difference in the wires, your power supply should have an indication. 

Most labels show the polarity of the pin and body of the plug:

Here's one for the Dustbuster power supply:



Here I shows the central pin as the positive terminal.
If there is no indication on the wire to help you will need a multimeter to ascertain which wire goes to the pin on your cable end ….and measure the two wires from the power supply to ascertain which one is positive. 

Most devices include reverse voltage protection since it's easy for users to use the wrong power supply, but it's anyone's guess as to whether your device is protected.
If you don't have a multimeter ….they are dirt cheap, so get one.