Interdentals in English: One or two phonemes?

The (inter)dental sounds dh and th in English are distinguished by all speakers, but simple rules with few exceptions can accurately predict which sound is used in any given word. Because grammatical category and some etymology are needed for the rules, this may not be enough reason to conclude that there is only one combined th-dh phoneme in English.
Yet there are very few minimally differing pairs.

The rules for prediction of th/dh, with examples, are as follows:

In initial position:

Voiceless [T]:
- Nouns: thing, thorn, thought, thus (meaning incense)
- Verbs: think, throw, thank
- Adjectives: thin, thick, thewy, theoretic
- Adverbs derived from adjectives by suffixing -ly: thinly, thoroughly
- Numerals
  - Cardinals: three, thirteen, thirty, thousand
  - Ordinals: third, thirteenth, thirtieth
- Prepositions: through
Voiced [D]:
- Adverbs (not made by suffixing -ly): thus, then, there
  (EXCEPTION: voiceless when r follows: thrice)
- Pronouns
  - Relative: that
  - Demonstrative: this, that, these, those
  - Personal: thee, they
  - Possessive: thy, thine, their, theirs
- Conjunctions: than, though, thence
- Definite article: the

In medial position:

Voiceless [T]:
- Words of Greek origin: sympathy, mathematics, author, epithet. But: rhythm (pronounced by some with [D]), vs. rhythmic!
- Biblical names, mostly of Hebrew or Greek origin: Methusalah, Golgotha, Bartholomew
- Proper names (some also because they're from Greek): Cynthia, Bertha, Dorothy, Timothy
- Compounds and suffixed words: something, nothing, strengthen, lengthen
  (EXCEPTIONS: worthy, swarthy, Smithy (also with [T]), but earthy, stealthy again with [T]).
Voiced [D]:
- elsewhere: father, farther, gather, brother, brethren, whether, rather.
  (EXCEPTION: brothel)
  (Note: brethren is one of the very rare cases where a voiced [D] and an [r] touch in the same word)

In final position:

Voiceless [T]:
- Nouns: moth, myth, length, growth, wreath, sheath
  (Note: many of these words' plurals have [Dz], at least with some speakers)
  (EXCEPTIONS: scythe, Hythe, booth, tithe)
- Adjectives: loath, uncouth (BUT: see under final [D] too).
- Ordinal numerals: eighth, seventh, fifth
Voiced [D]:
- Verbs: bathe, loathe, breathe, wreathe, sheathe
  (EXCEPTION: obsolete inflected forms: hath, doth, receiveth)
- Adjectives: lithe (BUT: see under final [T] too).
- Prepositions: with (though some say [wIT]; many others do that only in compounds: withhold, withdraw, forthwith)

Note 1:

Among the very few cases in which the difference between [D] and [T] distinguishes two otherwise identical words are:

loath	loathe
wreath	wreathe
sheath	sheathe
thus (incense)	thus
ether	either (Am. pronunciation)
thigh	thy
sooth	soothe
mouth (noun)	mouth (verb)
thou (short for 1000)	thou (pronoun)
teeth	teethe

All of these are, however, according to the rules mentioned above.

Note 2:

The rules for final [D]/[T] in nouns, adjectives and verbs also apply for [s]/[z] and [f]/[v] in some cases:

use (noun)	use (verb)
loose (adjective)	lose (verb)
life (noun)	live (verb)