The idea of being split into two can be broken down into different concepts
Good vs Evil
example: Good Ash vs Bad Ash in Army of Darkness.
Here, we have our character who is generally seen as inherently "good". The clone/twin is inherently evil. The evil character is often a "mirror" or complete opposite of the original hero.
example: Xander Harris in Buffy: The Vampire Slayer episode The Replacement.
Here, we have our character whose personality traits or qualities are pulled into separate beings. Instead of a character sometimes being confident or sometimes timid, Being A will always be confident while Being B will always be timid.
While these may seem awfully similar, there is a major difference: with trait/quality splits, the separate beings are just distilled traits and qualities. A character who is inherently good usually does not have a fully evil side, so their twin would not be evil.
So Where does that leave Po?
Po was split into two by The Mystical Mirror of Yin and Yang, which is described as
a mirror that possesses magical properties to separate its viewer into two beings — one representing the viewer's good nature, and the other their evil nature.
So, here we have the Good vs Evil split. Bad Po is the exact opposite of Good Po and is described as
As the physical manifestation of Po's evil nature, Bad Po was purely evil in all intentions. As described by Shifu, Bad Po possessed all the qualities and traits of regular Po's Dragon Warrior side (often used when he fights) — of such include aggression, indignation and impatience.
So, without the Good vs Evil split, Po never would have done any of those things. Nor would he have ever really even thought about them.