I would not say that you would have to do that. Researchers that I know and work that I have read in statistical literature regarding Likert scales claim that it is best to have questions reversed on occasion so that someone doesn't produce a biased result from laziness. You need to know which questions in your survey have that construct so that you can account for it, but that is it.
Here is a link to the paper I am talking about.
https://www.researchgate.net/profile/J_Barnette/publication/247728422_Effects_of_Stem_and_Likert_Response_Option_Reversals_on_Survey_Internal_Consistency_If_You_Feel_the_Need_There_is_a_Better_Alternative_to_Using_those_Negatively_Worded_Stems/links/546208030cf27487b4557a6e.pdf
Don't negatively word everything, but having some can be a good thing.
Article Effects of Stem and Likert Response Option Reversals on Surv...
The question does not have to be worded using negatives to result in opposite agree/disagree answers from other answers.
If in some questions "agree" signals a positive answer, and in a few "agree" signals a negative feeling, NOT reversing the coding could distort your overall averages.
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