ADHYATMA - roughly SPIRITUALITY
झन्क्रित कर संवेदना को
नर तू नारायण हो जा,
हिरण्यगर्भ की प्रणश्चेतना जगा
मन तू ब्रहमन हो जा,
कालातीत अस्तित्व है तेरा
जो जाएगा, फिर आएगा,
उद्गीत सुने जो अवचेतन में
शक्ति का वो शिव हो जा
~ Adiguru Prakriti
Adhyatma is not the same as Spirituality
Spirituality is defined as "the quality of being concerned with the human spirit or soul as opposed to material or physical things". While the word Spirit is defined as "the non-physical part of a person which is the seat of emotions and character; the soul". Now, you can also surf the internet and look for the meaning of Soul (which as per definition is not there in plants, animals and women, hence cannot be equated to Atma).
You will realize that all these definitions are talking about the Physical Existence of the Human/Being, the Atma. Even the Mind/Emotions are considered as parts of Physical Existence in Yogic Darshan (roughly philosophy).
The line that westerners draw between physical and non-physical is dependent upon what can be seen through eyes. While the line that Vedic or Yogic Sanskriti, people/seekers draw is dependent upon the "being/Atma" and all that the Atma has, she owns and that includes the physical body, 5 senses, mind, emotions, memory, intellect, logical/rational faculty, energy body, prana/pranic shakti, intuition/supernatural wisdom faculty, consciousness and the sheath of bliss.
Physical Body is Gross; while mind, emotions, memory, intellect, logical/rational faculty, energy body are Subtle; Prana, pranic shakti, intuition/supernatural wisdom faculty are even Subtler; Consciousness is even more, Subtler; Sheath of Bliss is further Subtler, and Atma is the Subtlest or Causal for herself. Paramatma/Divine is beyond the definitions of Gross and Subtle and hence is Causal for an Atma yet without any cause for himself/herself/itself.
The English language doesn't have words for the Subtle dimensions of life, of existence, hence most of the Sanskrit/Hindi words are incorrectly, roughly, loosely, approximately translated in English and hence lose the meaning in its entirety.
Adhtayma means "turning inward, turning away from all that is physical, subtle, subtler and turning towards Atma the Subtlest, the Real You, the Being". In other words, it means the "journey of an Atma" towards the "Realization" of the Ultimate Truth, Permanence. Which is -
There are 3 that exists eternally or are permanent or are Sat - Prakriti (Equilibrium of Sattva, Rajas, Tamas), Atma, Paramatma.
There are 2 that are Chitt/Conscious - Atma & Paramatma.
There is 1 that is Ananda - Paramatma.
Hence, Sat-Chitt-Ananda. The journey of an Atma for the realization of Sat-Chitt-Ananda through direct perception is Adhyatma. The net result of which is Moksha, ultimate freedom from the birth-death cycle.
So, you can see that Adhyatma is not the same as Spirituality, because of the way Westerners and Vedic/Yogic Darshana, Indian Rishis/Sages differentiates between what is Gross and what is Subtle and Causal.
Turning inward doesn't mean, turning towards your mind, emotions, memories, senses and so on. It means applying the techniques, the science of Yoga to find out who you truly are; an Atma and after Self Realization (Atma-Bodh/Samadhi/Enlightenment) living as an Atma, forever dwelling in the Ananda of Paramatma.