Forensic linguists use grammar, syntax and vocabulary to help crack cold cases

> In linguistics, reduplication is a morphological process in which the root or stem of a word, part of that or the whole word is repeated exactly or with a slight change.

The ablaut section also seems to suggest that "eat" should come before "have" anyway.

> In ablaut reduplications, the first vowel is almost always a high vowel or front vowel (typically ɪ as in hit) and the reduplicated vowel is a low vowel or back vowel (typically æ as in cat or ɒ as in top).

I suspect you feel that it flows more smoothly because it's more familiar. You have to stop a brain process that's become a bit automatic to say that phrase and instead say it slightly differently.

I would argue that it is, given the semantic shift that "to have" has undergone.

“To have” historically has had a more tangible sense of holding or owning something in a lasting, physical way. For example, in medieval and early modern English documents, “to have” frequently referred to holding physical property or goods in a manner implying true, ongoing possession. For instance, the formula “to have and to hold,” found in English property grants and other legal charters dating back to at least the 13th century, specifies that the grantee possesses the land not just in theory, but in continuing, tangible stewardship. This phrase does not simply mean ownership on paper—it affirms the right to keep and maintain the property indefinitely.

Today, "to have" is more abstract, and implies enjoying a condition or availability. In modern English we often use “to have” for intangible states, experiences, or conditions, rather than strictly physical possession. We say we “have time” or “have a headache,” meaning we experience or hold a certain condition, not that we own a concrete object. Saying “I have an idea” frames “idea” as something you possess, but it’s more about the existence of that thought rather than controlling a physical thing. We “have a meeting,” which implies an event scheduled for us to attend, not an object we keep. Over time, “to have” evolved to mark various states—emotional, temporal, conceptual—thus shedding some of its older, property-focused sense and becoming a flexible verb denoting conditions or availability.

So, because we interpret “to have” as less tied tangible possession, the original logic—that once you eat the cake, you cannot still "have it"— doesn’t strike the ear as sharply when we switch the word order.

I would suggest adding a time marker like “then” (e.g., “You cannot eat your cake and then have it, too.”) emphasizes the sequence and delineates that the action of eating precedes the attempt at possession.