Introduction

Some months are completely symmetric, meaning they have central symmetry as well as reflection symmetry, like February of 2010:

 February 2010
┌──┬──┬──┬──┬──┬──┬──┐ 
│ │ │ │ │ │ │ │ 
├──┼──┼──┼──┼──┼──┼──┤ 
│ │ │ │ │ │ │ │ 
├──┼──┼──┼──┼──┼──┼──┤ 
│ │ │ │ │ │ │ │ 
├──┼──┼──┼──┼──┼──┼──┤ 
│ │ │ │ │ │ │ │ 
└──┴──┴──┴──┴──┴──┴──┘ 

Some months have only central symmetry, like February of 1996 or current month, the April of 2018:

 February 1996
 ┌──┬──┬──┬──┐
 │ │ │ │ │
 ┌──┬──┬──┼──┼──┼──┼──┤
 │ │ │ │ │ │ │ │
 ├──┼──┼──┼──┼──┼──┼──┤
 │ │ │ │ │ │ │ │
 ├──┼──┼──┼──┼──┼──┼──┤
 │ │ │ │ │ │ │ │
 ├──┼──┼──┼──┼──┴──┴──┘
 │ │ │ │ │
 └──┴──┴──┴──┘
 April 2018 ┌──┐
 │ │
 ┌──┬──┬──┬──┬──┬──┼──┤
 │ │ │ │ │ │ │ │
 ├──┼──┼──┼──┼──┼──┼──┤
 │ │ │ │ │ │ │ │
 ├──┼──┼──┼──┼──┼──┼──┤
 │ │ │ │ │ │ │ │
 ├──┼──┼──┼──┼──┼──┼──┤
 │ │ │ │ │ │ │ │
 ├──┼──┴──┴──┴──┴──┴──┘
 │ │
 └──┘

And some are asymmetric, like the previous month, the March of 2018:

 March 2018
 ┌──┬──┬──┬──┐
 │ │ │ │ │
┌──┬──┬──┼──┼──┼──┼──┤
│ │ │ │ │ │ │ │
├──┼──┼──┼──┼──┼──┼──┤
│ │ │ │ │ │ │ │
├──┼──┼──┼──┼──┼──┼──┤
│ │ │ │ │ │ │ │
├──┼──┼──┼──┼──┼──┼──┘
│ │ │ │ │ │ │
└──┴──┴──┴──┴──┴──┘

Task

Take an input in form of a date, e.g.:

• 2018.04
• 2018.03
• 2010.02
• 1996.02

Output the corresponding symmetry, e.g.

• 2018.04 -> centrally symmetric
• 2018.03 -> asymmetric
• 2010.02 -> symmetric
• 1996.02 -> centrally symmetric

Rules

• This is code golf, so the smallest number of bytes wins.
• Standard loopholes are obviously not allowed.
• Assume that the week starts with Monday (thanks to Angs and Arnauld for suggestion).
• Consider only years between 1900 and 2100 (inclusive).
• The input and output formatting rules are permissive, meaning you can use any equivalent format that is native to the language of your choice.
• Base your solution on the Gregorian calendar.