Leaping Kangaroos

MATLAB, 92 90 86 84 bytes

n=input('');p=eye(n)+32;A=repmat([fliplr(p),p,''],1,2^n/2);A(:,n+1:n:end)=[];disp(A)

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eye creates an identity matrix. If we flip it and concatenate the original i.e. [fliplr(p),p] we get (for n=3):

0 0 1 1 0 0
0 1 0 0 1 0
1 0 0 0 0 1

With repmat(...,1,2^n/2) we repeat this 2^(n-1) times and get

0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0
0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 ...
1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1

From this we just delete the unnecessary columns, with A(:,n+1:n:end)=[];

Charcoal, 14 bytes

ＮλＰ^×λoＦ⁻λ¹‖Ｏ→

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Explanation

Ｎλ inputs an integer into λ. Ｐ^ is a multidirectional print (SE and SW) of ×λo (string multiplication of λ with o). Then Ｆ⁻λ¹ runs a for loop λ - 1 times, in which ‖Ｏ→ reflects the whole thing to the right with overlap.

05AB1E, 12 10 bytes

Îj¹FÐvû},À

Explanation:

Î              # Push zero and input
 j             # Prepend input - 1 spaces
  ¹F           # Input times do..
    Ð          #   Triplicate the string
     v }       #   Length times do..
      û        #     Palindromize
        ,      #   Pop and print with a newline
         À     #   Rotate the string on to the right

Uses the CP-1252 encoding. Try it online!