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05AB1E, 17 bytes

^{12 bytes if trailing whitespaces were allowed: try it online.}

ζε2úíJNúS}ζJðδÜõÜ

Port of @Shaggy's Japt answer.

Input as a list of list of characters; output as a list of lines.

Try it online.

More interesting answer using the Canvas builtin ((削除) 27 (削除ここまで) 25 bytes):
^{(削除) 20 (削除ここまで) 19 bytes if trailing whitespaces were allowed: try it online.}

€gDø4.ý ̃I€ûðoý374S.Λ¶¡ðδÜ

-2 bytes thanks to @emanresuA.

Input as a list of words; output as a list of lines.

Try it online.

Explanation:

ζ # Zip/transpose; swapping rows/columns of the (implicit) input-list of lists
 # of characters, with a space " " as filler for shorter lists
 ε # Map over each inner lists of characters:
 2ú # Pad each inner character with 2 leading spaces
 í # Reverse each inner string, so the spaces are trailing
 J # Join this list together to a single string
 Nú # Pad the 0-based map-index amount of leading spaces
 S # Convert the string back to a list of characters
 }ζ # After the map: Zip/transpose back again with " " filler
 J # Join each inner list together to a single string
 δ # Map over each inner line:
 ð Ü # Trim trailing spaces
 õÜ # Trim trailing empty strings from the list
 # (after which the list of lines is output implicitly)

€gDø4.ý ̃I€ûðoý374S.Λ¶¡ðδÜ
€g # Get the length of each word of the (implicit) input-list
 Dø # Convert each value to a pair of this value with a Duplicate + Zip/transpose
 4.ý # Intersperse this list of pairs with 4 as delimiter
 ̃ # Flatten it to a single list
I # Push the input-list of words again
 €û # Palindromize each word
 ðoý # Join this list with " " (2 spaces) delimiter
ŽDā # Push 374
 S # Convert it to a list of digits: [3,7,4]
.Λ # Use the modifiable Canvas builtin with these three arguments
 ¶¡ # Split it on newlines
 δ # Map over each inner line:
 ð Ü # Trim trailing spaces
 # (after which the list of lines is output implicitly)

Additional information about the Canvas builtin Λ/.Λ:

It takes 3 arguments to draw an ASCII shape:

1. Length of the lines we want to draw
2. Character/string to draw
3. The direction to draw in, where each digit represents a certain direction:

7 0 1
 ↖ ↑ ↗
6 ← X → 2
 ↙ ↓ ↘
5 4 3

€gDø4.ý ̃I€ûðoý374S creates the following Canvas arguments:

• Lengths (€gDø4.ý ̃): e.g. [11,11,4,7,7,4,3,3,4,4,4,4,4,4] for input ["programming","puzzles","and","code","golf"]
• Characters (I€ûðoý): e.g. "programmingnimmargorp puzzleselzzup andna codedoc golflog" for input ["programming","puzzles","and","code","golf"]
• Directions (374S): [3,7,4], which translates to \$[↘,↖,↓]\$

Because we have a list of lengths, those are leading here.

Step 1: Draw 11 characters ("programming") in direction 3 (\$↘\$):

p
 r
 o
 g
 r
 a
 m
 m
 i
 n
 g

Step 2: Draw 11-1 characters ("nimmargorp") in direction 7 (\$↖\$):

p
 r
 o
 g
 r
 a
 m
 m
 i
 n
 g

Step 3: Draw 4-1 characters (" p") in direction 4 (\$↓\$):

p
 r
p o
 g
 r
 a
 m
 m
 i
 n
 g

Step 4: Draw 7-1 characters ("uzzles") in direction 3 (\$↘\$):

p
 r
p o
 u g
 z r
 z a
 l m
 e m
 s i
 n
 g

etc.

See this 05AB1E tip of mine for an in-depth explanation of the Canvas builtin.

€gDø4.ý ̃ could alternatively be εgD4)} ̃ ̈ or €gDø4ドル¦ ̃, but I've been unable to find anything shorter.