Does it matter which way chain is installed after cleaning it?
 

After taking the chain off and cleaning it, I can't tell which direction my chain formerly faced. Does this matter?

nope

It depends.

Some chains are asymmetrical either having different plates on the inner (towards wheel) and outer side, or an inside (towards sprocket) or outside.

if your chain is symmetrical and you use a connector link, then it doesn't matter, though some theorize that you can squeeze more life out of it by reversing direction.

If you use a pin closure like Shimano or Campagnolo, check the instructions because some specify that the pin must be at specific end of the outer plate.

In addition, Shimano recommends that you install the new pin in the same direction that you pushed out the old pin.

I have the Sram connecting pin. Thanks for the help, folks.