Care And Pruning: Tips For Pruning A Yucca
The article also covers how to propagate more plants from the pruned material.

Here is a link that might be useful: Care And Pruning: Tips For Pruning A Yucca

better just to move it where it can assume the natural size and shape evolution has given it over millions of years. work with your plant's DNA rather than trying to fight it. both you and your plants will be happier.

I don't think this is a Yucca at all. The OP is in the UK (clues ... 'front garden' and something indefinable about the appearance of the street ;-) then I checked the member page) and the plant is pretty certainly Cordyline australis. It can grow to be a sizeable tree. Personally I don't find them attractive but they are pretty common in Southern UK and coastal areas. They can take a hard chop back but then you will not be able to predict where the new growth will come from. It could go off in any direction and the symmetry will be lost. Google up some images to find out how it will look in a few years time outside your upstairs window.

Here is a link that might be useful: Cordyline australis