A reed, sometimes seen adorning a cross, represents that held by Jesus just prior to His crucifixion. It was mockingly given to Him as a royal Sceptre and has since become part of the Arms of Christ.
Long before the Geneva Convention on Human Rights, which forbids any humiliating and degrading treatment of prisoners, when Jesus was crucified such practices were common. And when the victim had a high profile, well, it's not surprising the Roman soldiers wanted to enjoy the opportunity of showing how tough they were.
(Has civilisation progressed much?)
After scourging Jesus, the soldiers placed a scarlet robe over His shoulders and a crown of thorns on His head. To complete the regal image, they put a rod in His right hand as a sceptre, then with a mocking bow cried out, "Hail, King of the Jews!" (Matthew 27:29.)
It is likely that the rod was a reed baton, similar to sticks used by security forces today for controlling unruly crowds. In the next verse (Matthew 27:30), we read how the soldiers then took the rod and beat Jesus about the head, no doubt pushing the thorns further into His scalp.
The soldiers were, of course, absolutely correct is addressing Jesus as King. Unbeknownst to them, they were also fulfilling a prophecy of Jesus death (Matthew 20:17-19).
The sceptre can also be viewed as a shepherd's staff (see Shepherd's Crook and Cross).
For other another Reed Cross, see St. Brighid's Cross.