Choosing between Regression and Classification
If you are trying to build a neural network you should stick with a the orientation as a numeric value since neural networks only accept numeric data as input.  If you want to use the class values, you will need to one hot encode them.  It will also result in a loss of information.

Of course, it depends what kind of neural network you are trying to implement. In most cases, I don't think this step is necessary and I think it may complicate the architecture of your model.

If you are open to suggestions, instead of building a regression model, I would investigate Principal Component Analysis.  It could alleviate the need to have the photographs labeled before input into the neural network.