The Swastika and the cross have the same origin.
I'm not sure what you meant by that statement. I can't speak to the origins of the Swastika before Hitler appropriated it for his own uses. But I can speak a little bit to the cross.
The cross does have in common with the Swastika that it had a different origin that that for which it is most well known today, meaning that both are symbols appropriated by later groups who in essence changed their original meaning so that their present meaning has nothing to do with their original usage.
Originally, the cross was simply an instrument of torture. There was no symbolic meaning to the Roman's cross at all. Though their were other symbols looking very similar to a cross, such as an Egyptian ankh, there is no actual connection between the cross that Romans used as an instrument or torture and the Egyptian symbol which was originally a simple heiroglyph.
Romans did execution by crucifixion on a number of different structures. Sometimes just a single stake in the ground. Sometimes on a whole lattice frame with mutliple interwoven uprights and cross bars. And sometimes in a single upright with a single crossbar. In the latter case it was common for the uprights to be permament erected and for the condemned man to be asked to carry his own crossbar. According to the Gospel accounts, this appears to be the case with Jesus, so it gives us our best hint as to what the cross that Jesus was crucified on actually looked like.
Later, despite the reality that death by crucifixion was so horrific and scandalous that Roman citizens who were sentenced to death could request decapitation in order to avoid crucifixion, and that execution by crucifixion was generally reserved for the most heinous acts which threatened the state, usually crimes of sedition and insurrection -- despite all of this, early Christians continued to emphasize Jesus death on the cross as a key point of Christian theology. It expressed their belief in the grace of God and the atonement from sin which was realized by Jesus' sacrifice on a cross. For this reason the cross which had initially been understood only as an instrument of human torture became a symbol of divine love.
In time, as with all symbols, the symbol itself evolved taking on many forms. People would see it in the shape of an anchor, hence evolved the Anchor Cross. Others would see it in the four points of a compass or the letter "T". And so other shapes and patterns became accepted for a variety or different reasons, but the reason they were adopted was not to promote the original symbol, but because of the importance the story of the cross had developed within the theology of the Christian Church.
