"-"  has been used to set the reference in the global environment so as to avoid any scoping issues in recursive calls.

# Palindrome Calling Function  

 palindrome <- function(astr) {
   
  # house keepging removing blanks and punctuation etc through a pattern
  # make all to lower case 
 
  regex <- "^[ \t]+|[ \t]+$|\\.|\\'|[ \t]+|,|!|-"
  astr.clean <- gsub(regex,'',astr)
  str.refined <- tolower(astr.clean)
  shortstring <- palindromecal(str.refined)
  return(shortstring)
}

Test cases are shown for reference:
# Test1 :: True

palindrome('aa')
# > palindrome('aa')
# [1] TRUE
palindrome('A but tuba.')
# > palindrome('A but tuba.')
# [1] TRUE
palindrome('A man, a plan, a cat, a ham, a yak, a yam, a hat, a canal-Panama!')
# > palindrome('A man, a plan, a cat, a ham, a yak, a yam, a hat, a canal-Panama!')
# [1] TRUE
palindrome(' ')
# > palindrome(' ')
# [1] TRUE
palindrome('aaaabbaaaa')
# > palindrome('aaaabbaaaa')

# Test2 :: False

palindrome('..a.....b')
# > palindrome('..a.....b')
# [1] FALSE
palindrome('abcdefghijklmnopqrstuvwxyz')
# > palindrome('abcdefghijklmnopqrstuvwxyz')
# [1] FALSE

Lastly any R discussion would be incomplete without a reference to the vectorization, this function can be used with other standard vector functions such as "lapply and sapply".

list.string <- list('abc',"aa",'aaa','ab')
list.new  <- lapply(list.string,palindrome)
vec.new  <- sapply(list.string,palindrome)