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#TI-Basic (TI-84 Plus CE), 131 bytes

Prompt Str0
inString(Str0," →A
length(Str0)–A→L
sub(Str0,A,L→Str0
" →Str2
For(B,1,L
sub(Str0,B,1)+Str2→Str2
End
inString(Str2," →A
L-A→L
sub(Str2,A,L→Str2
" →Str0
For(B,1,L
sub(Str2,B,1)+Str0→Str0
End
Disp Str0

Explanation:

Prompt Str0              # 4 bytes, input user string
inString(Str0," →A       # 10 bytes, store the index of the first space in A
length(Str0)–A→L         # 10 bytes, store the length of the string (after removing the first A characters) in L
sub(Str0,A,L→Str0        # 12 bytes, remove first A-1 and last characters of Str0
" →Str2                  # 6 bytes, temporary string to reverse Str0
For(B,1,L                # 7 bytes, reverse Str0 into Str2
sub(Str0,B,1)+Str2→Str2  # 16 bytes
End                      # 2 bytes
inString(Str2," →A       # 10 bytes, store the index of the first space in A
L-A→L                    # 6 bytes, get new length
sub(Str2,A,L→Str2        # 12 bytesremove first A-1 and last characters of Str2
" →Str0                  # 6 bytes, temporary string to reverse Str2
For(B,1,L                # 7 bytes, reverse Str2 into Str0
sub(Str2,B,1)+Str0→Str0  # 16 bytes
End                      # 2 bytes
Disp Str0                # 3 bytes, display final string with (2 trailing spaces)