It appears that this question is one of terminology, so I am answering it as such.
Convention for representing features in DNA sequences
The convention is that in indicating any sequence feature† in a protein-coding gene on double-stranded DNA, a single strand‡ is represented — the one from which the amino sequence could be read using the genetic code (conceptually, with T substituted for U). Like any other nucleic acid sequence§, it is always written in the 5ʹ to 3ʹ direction in the same manner as the mRNA transcribed from it, without this being explicitly stated.
† An exception might be hemi-methylation, in which case both strands would be shown.
‡ I call this the sense strand. I discuss the nomenclature further below.
§ An exception is that tRNA anticodons are sometimes written in the 3ʹ to 5ʹ direction for ease of comparison with the codon, but in this case the directionality is indicated.
Origin and justification for this convention
Historical. The amino acid sequence of the protein (the product of the gene) is central to this convention because knowledge of the genetic code, and hence representation of the region of the mRNA that encodes protein — and by extension the DNA — was the first sequence information to be known.
Logical consistency. Later other sequences features were identified (some of which initially may have just been genetic features), e.g. ribosome binding sites, polyadenylation addition signals, transcription start sites, promoters, transcription factor recognition sites. It was logically consistent to represent them on the same strand as the coding sequence.
Functional agnosticism. In many cases the function of a sequence followed its description, so there was no reason initially to place it on any particular strand. However, even if it were thought that the function of some sequence were to be recognized on the opposite strand (what I would call the anti-sense strand), it would be unwise scientifically to change the representation to indicate this. Science progresses and interpretation changes. Better to separate concrete descriptive features from conclusions about their function.
It couldn’t ever be an ATAT box
Even if you represented the TATA box on the anti-sense strand, it could never be called an ‘ATAT box’ (as suggested by the poster) because, according to the basic convention, ATAT is 5ʹ-ATAT-3ʹ, and on the antisense strand the sequence is 3ʹ-ATAT-5ʹ, i.e. TATA!
Terminology for referring to the two strands of dsDNA
Whereas the above is the convention followed universally, the following is just my opinion. In science terminology is important to communicates ideas unambiguously, so I think it worthwhile explaining the ambiguity in some of the terms, the use of which I discourage.
Sense and anti-sense
This is my preferred term because, although not perfect, it avoids the pitfalls of the others. The idea seems to me clear that when you read the string of codons that encode the amino acid sequence from this strand they ‘make sense’. (Anti-sense is used in preference to non-sense, as ‘nonsense’ was the term used historically for mutations that converted amino acid codons into stop codons.) It can also be extended to non-protein coding genes (e.g. for tRNA), where ‘sense’ correlates with the sequence of the gene product.
I shall use ‘sense’ and ‘anti-sense’ as reference terminology in discussing other terms.
Coding and non-coding
This has the disadvantage that it cannot be extended to non-protein coding genes. However my primary objection is that it can cause confusion, as coding only unambiguously refers to mRNA. As the anti-sense strand is the template for RNA polymerase, one might make the mental association between this and ‘coding’, whereas it is the sense strand that is meant in this (admittedly common) usage.
Template and non-template
Template might be a more logical term for the anti-sense strand as this is the template for transcription by the RNA polymerase (although mRNA is also a template — for translation). However it is used only infrequently.
Plus and minus
This terminology is used for single-stranded (especially RNA) viruses to represent the whole genome, where in plus-strand viruses the genome is also the mRNA, i.e. the sense strand. One problem here is that it can be confusing for the beginner, who by extrapolation may assume that in double-stranded DNA genomes one strand is the sense strand for all genes. It is not. Which brings me to my final point…
…whatever terminology you use, it is better to make sure it is clear that you are referring to the sense or anti-sense strand of a gene, not of the whole genome. You need to employ some other terminology to distinguish between the two strands of e.g. bacterial or plasmid DNA if it is necessary to distinguish them.
