Theory:

Macavity’s a ginger cat, he’s very tall and thin;
You would know him if you saw him, for his eyes are sunken in.
His brow is deeply lined with thought, his head is highly domed;
His coat is dusty from neglect, his whiskers are uncombed.
He sways his head from side to side, with movements like a snake;
And when you think he’s half asleep, he’s always wide awake.
Explanation:
 
The third stanza focuses on the cat's physical appearance. We learn a lot more about Macavity in the stanza.

Macavity is a ginger cat. He is so-called for the colour of his fur: orangish-brown. He is tall and skinny, and his eyes are sunken in. His eyebrows are deeply creased with thinking, and his head is considerably arched. His fur looks dusty as Macavity rarely pays attention to it, and his whiskers are unkempt.
 
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Macavity is a tall and thin ginger cat

The above description of Macavity tells us that Macavity is not bothered about keeping or maintaining his appearances. The lines also reveal that Macavity is probably a street cat.
 
Like any other cat, Macavity tends to swing his head from side to side like a snake. And even though he appears to be sleeping or half-asleep, he is always wide awake. The line shows that Macavity is always on alert and explains his ability to remain hidden from the Scotland Yard.
 
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Though it may seem like he is half-asleep, he is always wide awake
 
The phrase “with movements like a snake” is an example of simile as the movement of the cat’s head is compared to that of a snake.
 
Meanings of difficult words:
 
S.No
Words
Meanings
1
Sunken (Of eyes or cheeks) Having curved or moved inwards
2
DomedShaped like a rounded arch
3
SwaysTo move slowly or rhythmically backwards and forwards or from side to side
4
Whiskers A long projecting hair or bristle growing from the face or snout of many mammals
Reference:
National Council of Educational Research and Training (2008). Honeydew. Macavity: The Mystery Cat - T. S. Eliot (pp. 50-51). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.