The best option is to normalize all Likert variables, including those with 5-points. You can find the option in Descriptive statistics in SPSS. If I am correct: Analyze - Descriptives - ... And one of the first three. Tick the box 'save as standardized values' (z scores). All variables will be transformed in a way that they will be ranged from -1 to 1.
Why do you have to change the scoring? This could affect the reliability and validity of the scale, as well as your ability to compare its results to earlier versions of the scale.
Most measures of reliability (e.g., Cronbach's alpha) and validity (e.g., convergent validity) are based on correlations, so they are already inherently normalized.