What is the scientific term for the erroneous inversion of cause and effect?

Would the concept of contraposition from the world of mathematical logic work for you?

From the Wikipedia explanation of contraposition:

The contrapositive of a statement has its antecedent and consequent inverted and flipped. For instance, the contrapositive of the conditional statement "If it is raining, then I wear my coat" is the statement "If I don't wear my coat, then it isn't raining."

Given your comment to the first answer...

The logical error I am looking for, is "I am wearing my coat, therefore it is raining"

...I think you are referring to deductive arguments. A deductive argument (definition taken from Mark Storey, Bellevue College):

...claims (explicitly or implicitly) that if the premises all are true, then the conclusion must be true

Valid examples are (taken from the linked document from Mark Storey)

• All bats are cute animals. No cute animals are mean. So, certainly, no bats are mean.
• Joan is Lauren’s mother. Therefore, Joan must be older than Lauren.
• Nobody knows Ned. Therefore, it must be that Ned does not know himself.
• Some cats are pets. Thus, some pets must be cats.

A familiar invalid example is the following (taken from memory on my own Philosophy class, a long, long time ago):

All cows have 4 legs. It has 4 legs, thus it must be a cow.

Your question proper states that it should be related to cause and effect. In terms of deductive reasoning, however, it is based on premises and conclusions. Quoting Mark Storey again:

[T]he argument may fail in its aim; what makes the argument deductive is that it is the kind of argument that—if it were successful—would have the premises absolutely guarantee the conclusion to be true.

I think what you are after are examples where the premises do not guarantee the conclusion to be true. This is different from causes and effects, though.

According to a dictionary of psychology this is simply called "reverse causality"

In attempting to understand the relationship between cause and effect, a tendency to attribute what is actually the cause to the effect. For example, does the ingestion of lead paint cause a lower intelligence quotient (IQ) or is it that children with lower IQs tend to eat lead paint?